Outside the high school program / Continuation and limits 64 exercises corrected

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1. Limits of sequences E.43 the sequence ( u n ) is defined on N by giving its first term u 0 =800 and the recurrence relation: u n +1 =0.6 · u n +400 1 Calculate u 1 and u 2 . 2 We define another sequence ( v n ) on N by posing for any natural number n : v n =1000 − u n . a Calculate the first three terms of this sequence ( v n ) . b Show that this sequence ( v n ) is geometric and deduce the expression of v n as a function of n . 3 a Deduce from previous results that : u n = 1000 − 200 × (0.6) n b What is the limit of the sequence ( u n ) ? E.59 Consider the sequence ( u n ) defined by u 0 =8 and for any natural number n : u n +1 = 1 2 · u n − 5 . 1 a Calculate the terms u 1 and u 2 b Is the sequence ( u n ) arithmetic? geometric? Answers will be justified. 2 Consider the sequence ( v n ) defined for any natural num-ber n by: v n = u n +10 . a Show that the sequence ( v n ) is a geometric sequence of reason 1 2 and calculate the first term v 0 . b Express the general term v n as a function of n . 3 Determine the limit of the sequence ( v n ) and then that of the sequence ( u n ) E.42 lebanon - June 2002 - 6 points -exercise of your choice Part A : study of a sequence The sequence ( u n ) is defined on N by giving its first term u 0 =800 and the recurrence relation: u n +1 = 0.6 · u n + 400 . 1 Calculate u 1 and u 2 . 2 We define another sequence ( v n ) on N by posing for any natural number n ,: v n =1000 − u n . a Calculate the first three terms of this sequence ( v n ) . b Show that this sequence ( v n ) is geometric and deduce the expression of v n as a function of n . 3 a Deduce from previous results that : u n = 1000 − 200 × (0.6) n . b What is the limit of the sequence ( u n ) ? Part B: application The president of a cultural association notes that each year the association keeps 60 % of its former members and that 400 new people join. It is assumed that these figures remain the same over the years. When this association was founded in January 2001 , there were 800 members. 1 Calculate the number of members in January 2002 . 2 What will be the number of members in January 2003 ? 3 In which year will the association’s membership exceed 990 for the first time? E.86 Let be the sequence ( u n ) defined by its first term u 0 =6 and by the recurrence relation: u n +1 = 1 4 · u n + 3 ( n natural number) 1 We pose : v n = u n − 4 . a Show that the sequence ( v n ) is a geometric sequence whose reason and first term will be specified. b Show that v n =2 · 1 4 n . Deduce the expression of u n as a function of n . c Determine the limit of the sequence ( v n ) , then that of the sequence ( u n ) 2 We pose : a n =ln v n a Show that the sequence ( a n ) is an arithmetic sequence of reason − 2 · ln 2 . b Determine the expression of a n as a function of n . c Determine the value of n for which a n is equal to − 13 · ln 2 . https://chingmath.fr chapExoCorrec/43 sacados/43 Liban - 2002 - au choix - 6 points chapExoCorrec/59 sacados/59 Antilles - 2002 - 6 points - obligatoire chapExoCorrec/42 sacados/42 Liban - Juin 2002 - 6 points - au choix sacados/86
E.1891 A miniature robot moves along a circle with center O and radius 1 meter. It is placed at point A 0 and then travels along the circle, turning in the forward direction (direction of the arrow in the figure) . It is decided that its journey will be carried out in several stages (see figure) . 1 Stage 1 : it travels along the semicircle from A 0 to A 1 , the distance traveled is noted as d 1 . 2 Stage 2 : it travels the quarter circle from A 1 to A 2 , the distance traveled is noted as d 2 . 3 Step n , with n 2 : it travels the arc of the circle from A n − 1 to A n , the distance traveled, noted as d n , being half that traveled in the previous step. Reminders: The perimeter of a circle with a radius of 1 meter is equal to 2 ı meters. 1 + q + q 2 + · · · + q n = 1 − q n +1 1 − q , for any real number q =1 . 1 a Express d 1 and d 2 in terms of ı . b Reproduce the figure, then place the point A 3 . Justify that : d 3 = ı 4 . 2 a What is the nature of the sequence ( d n ) ? Justify your answer. b Show that, for any integer n 1 : d n = ı 1 2 n − 1 . 3 We are interested in the total distance, denoted by D n , traveled by the robot on the circle at the end of step n a Calculate D 2 . b Show that, for any integer n 1 : D n = 2 ı 1 − 1 2 n . c Find the limit of the sequence ( D n ) . d Justify that, for any integer n 1 : D n < 2 ı What can we deduce about the robot’s movement on the circle? E.1889 The sales department of a news-paper has found that every year it registers 1 000 new sub-scribers, but 50 % or so of the old subscribers don’t renew. The aim of this exercise is to study the evolution of the num-ber of subscribers if this situation persists, knowing that in the past year, the newspaper had 4 000 subscribers. For this purpose, consider the sequence ( U n ) defined by: u 0 =4 and, for any natural number n : u n +1 =0.5 u n +1 1 Explain why, for any integer n 0 , u n is an approxima-tion of the number of thousands of subscribers after n years. 2 Reproduce and complete the table below : n 0 1 2 3 4 5 u n 3 Using reasoning by recurrence, show that : For any natural number n : u n > 2 4 Let ( v n ) be the sequence defined on N by: v n = u n − 2 . a Show that the sequence ( v n ) is a geometric sequence of reason 0.5. Specify the value of v 0 . b Deduce the expression of v n as a function of n . 5 a Using the result from the previous question, show that : For any natural number n : u n =2 · 1+0.5 n b What is the limit of the sequence ( u n ) ? c Give an interpretation of this limit. 2. Sum of terms E.56 The unit is the centimeter Consider an isosceles right triangle A 0 B 0 C 0 with side A 0 such that : A 0 B 0 = A 0 C 0 =20 . 1 Draw a figure that you will complete as you work through the exercise. 2 Let A 1 be the midpoint of segment [ B 0 C 0 ] , B 1 the mid-point of segment [ A 0 C 0 ] , and C 1 the midpoint of segment [ A 0 B 0 ] . a Draw triangle A 1 B 1 C 1 , which is an isosceles right tri-angle with right angle at A 1 . b Show that triangle A 1 B 1 C 1 is an isosceles right trian-gle with the right angle at A 1 . c Calculate the area of A 1 B 1 C 1 , denoted by u 1 , and the area of A 0 B 0 C 0 , denoted by u 0 . d Express u 1 in terms of u 0 . 3 Let A 2 be the midpoint of the segment [ B 1 C 1 ] , B 2 the midpoint of the segment [ A 1 C 1 ] , and C 2 the midpoint of https://chingmath.fr chapExoCorrec/1891 sacados/1891 A0A1A2O chapExoCorrec/1889 sacados/1889 chapExoCorrec/56 sacados/56 Japon - 2003 - 7 points - obligatoire
[ A 1 B 1 ] . a Calculate the area of triangle A 2 B 2 C 2 , denoted by u 2 . b Express u 2 in terms of u 0 . 4 By repeating this construction, we define a right isosceles triangle A n B n C n at A n . Let u n denote the area of this triangle A n B n C n . a Express u n in terms of u n − 1 and show that for any natural number n , we have : u n = 1 4 n × 200 . b Find the smallest integer p such that u p < 10 − 3 c Express the sum u 0 + u 1 + · · · + u n in terms of n . Use this to find the limit of this sum as n approaches + ∞ . Reminder : Arithmetic sequences with first term u 0 and common difference a : u n +1 = u n + a ; u n = u 0 + n · a 1+2+ ··· + n = n ( n +1) 2 Geometric sequences with first term u 0 and common ratio b : u n +1 = b · u n ; u n = u 0 · b n If b =1 : 1+ b + b 2 + b 3 + ··· + b n = 1 − b n +1 1 − b If b =1 : 1+ b + b 2 + b 3 + ··· + b n = n +1 E.31 The population of a city is growing steadily by 10 % per year. In 2000, it was 8 000 inhabitants. 1 Let u n denote the estimated theoretical number of res-idents for the year (2000+ n ) . We therefore have u 0 = 8000 . a Calculate the terms u 1 and u 2 . b Express the term u n +1 in terms of u n . Use this to derive the expression for the term u n in terms of n . c Calculate the projected population for the year 2006, rounded to the nearest whole number. d Determine in which year the population will have dou-bled 2 Let v n denote the increase in the number of inhabitants observed in year (2000+ n ) compared to the previous year. Therefore, for any nonzero natural number n , we have : v n = u n − u n − 1 . a Calculate the terms v 1 and v 2 . b Express the general term v n in terms of n . c Compute the sum v 1 + v 2 + ··· + v n in terms of n . Ver-ify, for the special case where n equals 6, the result obtained in 1 c E.45 1 Consider an arithmetic sequence ( u n ) of first term 20 and reason − 1 . a Express u n in terms of n . b Give the expression for the sum of the first n terms. c Deduce the following limit: lim n ↦→ + ∞ u 0 + u 1 + u 2 + · · · + u n 2 Consider a geometric sequence ( v n ) of first term 20 and reason 0.6 . a Express v n in terms of n . b Give the expression for the sum of the first n terms of the sequence ( v n ) . c Deduce the following limit: lim n ↦→ + ∞ v 0 + v 1 + v 2 + · · · + v n E.60 Gaston is hesitating between two employment contracts for which the first month’s salary is 1600 euros. Contract n o 1 : each month from the second month the monthly salary increases by 10 euros. Contract n o 2 : each month from the second month on-wards the wage increases by 0.6% compared with the previous month. 1 For each of the two contracts, determine the nature of the sequence of monthly salaries, specify the first term and the reason. 2 For each of the two contracts, calculate the total wages received during the first year. 3 Using the calculator, determine from which month on-wards the monthly salary corresponding to the n o 2 con-tract becomes higher than that of the n o 1 contract. Cor-rectly justify the answer. We recall: The sum S of the n first terms of a geometric sequence ( u n ) of reason q ( q =1) is : S = u 1 + u 2 + · · · + u n = u 0 · 1 − q n 1 − q The sum S of the first n terms of an arithmetic se-quence ( v n ) of reason r : S = v 1 + v 2 + · · · + v n = n · v 1 + r · n ( n − 1) 2 https://chingmath.fr chapExoCorrec/31 sacados/31 Antilles - 2003 - au choix - 6 points chapExoCorrec/45 sacados/45 chapExoCorrec/60 sacados/60
E.66 A globetrotter has bet to cover 5 000 km on foot. He can, fresh and ready, cover 50 km in a day, but every day fatigue builds up and so his performance decreases by 1 % every day. We’ll note d n the distance covered during the n -th day. 1 Calculate the distances d 1 , d 2 , d 3 covered during the first three days. 2 Explain why: d n +1 = 0.99 · d n . Deduce the nature of the sequence ( d n ) and the expres-sion of d n as a function of n . a Calculate, as a function of n , the total number L n of kilometers traveled after n days. L n = d 1 + d 2 + · · · + d n b Deduce the limit of L n when n tends to + ∞ . Can the globetrotter win? 3 Using the calculator, determine the minimum number of days N it would take him to travel 4 999 km We recall that : The sum S of the n first terms of an arithmetic se-quence ( u n ) of reason r is : S = u 1 + u 2 + · · · + u n = n · u 1 + u n 2 The sum S of the first n terms of a geometric sequence ( v n ) of reason q ( q = 1 ) is : S = v 1 + v 2 + · · · + v n = v 1 · 1 − q n 1 − q E.34 Part A : An artificial water reservoir is fed by a stream whose flow de-creases by 20 % from one day to the next due to the heat. For the day of 1 er June the flow D 0 is equal to 300 m 3 per day. 1 Calculate flow D 1 for June 2. 2 Let D n be the flow rate for n ième day after 1 er June. Ex-press D n +1 as a function of D n . What is the nature of the sequence ( D n ) ? 3 Express D n as a function of n . Calculate the flow D 29 for the day of June 30. Round to the nearest tenth of a cubic metre. 4 Calculate the volume of water brought into the reservoir over the 30 days of June. The result will be rounded to the cubic metre. Recall that : 1+ b + ··· + b n = 1 − b n +1 1 − b Part B: From July 1 er , the stream flow can be considered zero (less than 0.5 m 3 = jour ) . The heat in the reservoir causes 4 % of the total volume of water to evaporate per day. In addition, 500 m 3 of water must be released from the reser-voir every evening, after evaporation, due to drought. On the morning of 1 er July, the reservoir contains V 0 = 100 000 m 3 of water. 1 Let V n be the volume of water at n ième morning after 1 er July. a Show that V 1 is equal to 95 500 m 3 . b Express V n +1 as a function of V n . 2 Consider the sequence of general term U n defined for any integer n by: U n = V n +12 500 . Show that the sequence ( U n ) is a geometric sequence of reason 0.96 whose first term U 0 will be calculated. 3 Express the general term U n as a function of n . Deduce the expression of V n as a function of n . 4 Calculate V 31 the volume remaining on the morning of 1 er August. (The result will be rounded to the cubic metre) 5 At what date will the reservoir be at ˇ sec ı? E.1838 Let a be a real number, with a =1 . Show by recurrence that for any integer n ∈ N , we have : 1 + a + a 2 + · + a n = 1 − a n +1 1 − a https://chingmath.fr chapExoCorrec/66 sacados/66 chapExoCorrec/34 sacados/34 Japon - 2004 - Obligatoire - 7 points sacados/1838
E.44 In this exercise, the two parts are independent. The required results will be rounded to the nearest hundredth. To make a purchase costing 1 600 e , , a customer can choose between two payment methods. We will assume that the final payment may exceed the pur-chase amount, including credit fees. Part A In this problem, the first payment is 150 e , and a sequence of payments denoted by ( a n ) satisfies a 0 =150 and, for any integer n : a n +1 = 0 ; 95 · a n + 100 1 Calculate the second payment a 1 and the third a 2 . 2 Show that with five payments, the sum of 1 600 e is re-paid. Part B In this problem, the first payment is 200 e , , and each subse-quent payment is equal to the previous one minus 5% . Let ( b n ) denote the sequence of payments with b 0 =200 . 1 Verify that b 1 =190 and calculate b 2 and b 3 . 2 Express the term b n +1 in terms of b n . Determine the nature of the sequence ( b n ) and give the general term b n in terms of n . 3 Based on n , calculate the sum S n equal to b 0 + b 1 + · · · + b n of the ( n +1) first payments. Calculate the sums S 8 and S 9 and interpret the result. E.46 Let OA 0 A 1 be a right-angled trian-gle isosceles at A 1 . Externally to triangle OA 0 A 1 , we construct on side [ OA 1 ] a triangle OA 1 A 2 rectangle isosceles in A 2 . 1 We pose OA 0 =1 . Show that : OA 1 = A 0 A 1 = 1 2 2 We iterate the process and draw a sequence of isosceles right-angled triangles. If OA n − 1 A n is an isosceles right-angled triangle at A n , we then trace externally to it, the triangle OA n A n +1 rectangle isosceles in A n +1 . Establish that : OA n +1 = 1 2 OA n In the rest of the exercise, we note ( u n ) , the sequence such that u 0 = OA 0 , u 1 = OA 1 , . . . , u n = OA n 3 a Deduce from the previous question that the sequence ( u n ) is a geometric sequence and characterize it. b Note L n the length of the broken line OA 0 A 1 · · · A n − 1 A n , i.e.: L n = OA 0 + A 0 A 1 + ··· + A n − 1 A n = u 0 + u 1 + ··· + u n Establish that : L n = 1 − 1 2 n +1 1 − 1 2 . Deduce the limit of the sequence ( L n ) when n tends to + ∞ Formulaire If 0 <b< 1 then lim n → + ∞ b n =0 If ( v n ) is a geometric sequence with reason q ( q =1 ) and first term v 0 , then : S n = v 0 + v 1 + · · · + v n = v 0 1 − q n 1 − q https://chingmath.fr chapExoCorrec/44 sacados/44 Antilles - 2002 - 3 points - au choix chapExoCorrec/46 sacados/46
E.53 Urbain and Valérie earned the same degree in the same year and were both hired on 1 er January 2000 by two different companies under two different open-ended contracts : In the first year, Urbain’s net annual salary is 14 000 e , and this salary increases by 4 % each year on 1 er January compared to the previous year’s salary. Valérie starts with a net annual salary of 14 500 e , and this salary increases by 500 e each year on 1 er January. Let u n be Urbain’s net annual salary in euros for the year (2000+ n ) and v n be Valérie’s. With these notations, we have : u 0 =14 000 ; v 0 =14 500 . 1 a Calculate the terms u 1 and u 2 . b Show that the sequence ( u n ) is a geometric sequence, and determine its common ratio c Express the general term u n in terms of n . 2 a Calculate the terms v 1 and v 2 . b Show that ( v n ) is an arithmetic sequence, and state its common difference. c Express the general term v n in terms of n . 3 Calculate u 0 + u 1 + ··· + u 8 and v 0 + v 1 + ··· + v 8 . Interpret these results. Is the situation the same at the end of the following year? E.1837 1 Develop : ( n + 1)( n + 2) . 2 Show by recurrence that for any n ∈ N , we have : 1 + 2 + 3 + · · · + n = n ( n + 1) 2 E.4627 A figure is constructed successively as follows : We start with a square whose side measures 4 cm ; In the next step, we add a new square whose side mea-sures half the previously added square. Here are the first three steps of this construction represented : We note u n the total area of the figure constructed in step n : we have just constructed the sequence u n de real numbers. 1 Note ( v n ) la the sequence of areas of the squares added at each step of this construction. a Give the first five terms of the sequence v n . b What is the nature of the sequence v n ? c Give the explicit formula associating each term of the sequence v n en as a function of its rank. 2 a Determine the value of the first five terms of this sequence. b Show that the sequence u n vérifie the recurrence re-lation: u n +1 = u n + 4 · 1 4 n pour all n ∈ N 3 The explicit formula defining the sequence u n est of the form : u n = ¸ · 1 − ˛ n +1 ¸ ∈ R et ˛ ∈ R . Determine the value of ¸ and ˛ . https://chingmath.fr chapExoCorrec/53 sacados/53 Japon - 2002 - 5 points - au choix sacados/1837 chapExoCorrec/4627 sacados/4627 Etape no1 Etape no2 Etape no3
E.54 It’s rather curious that an infinite number of positive terms that you add as you go along can give a finite result. Thus the Greek Zeno claimed, in the IV ème century BC J . C . , demonstrate that it is impossible to go from one point to another, because ˇ before reaching the goal, one must reach the middle of the road, then reach the middle of the path to be traveled, and so on : as there are an infinite number of stages to observe, one cannot reach the end of one’s journey ı. Numbers and their mysteries - A. Warusfel I- Construction of the figure Construct a segment [ AB ] then, 1 the middle A 0 of [ AB ] , 2 the middle A 1 of [ A 0 B ] , 3 the middle of A 2 of [ A 1 B ] . II- Using a numerical sequence: We thus construct a sequence of points A n such that for any n integer greater than or equal to 1, A n is the midpoint of the segment [ A n − 1 B ] . It is assumed that AB =2 . We pose : d 0 = AA 0 ; d 1 = A 0 A 1 ; d 2 = A 1 A 2 and for any integer n 1 : d n = A n − 1 A n . 1 We have d 0 =1 , calculate d 1 and d 2 . 2 We admit that, for any natural number n : d n +1 = 1 2 d n . a Deduce that the sequence ( d n ) is a geometric sequence whose first term and reason will be specified. b Give the expression of d n as a function of n . 3 We pose : S n = d 0 + d 1 + d 2 + ··· + d n . a Check that : S n =2 · 1 − 1 2 n +1 . b What is the limit of S n when n tends to infinity? c Give a geometric interpretation. Form : Sum of the n +1 first terms of a geometric sequence of first term u 0 and reason q (with q =1 ) : S n = u 0 + u 1 + u 2 + · · · + u n = u 0 · 1 − q n +1 1 − q E.30 A landscape gardener needs to create a spiral planted with shrubs. He wants to know the length of this spiral in order to estimate the number of shrubs to be planted. Here’s the diagram he draws up : This spiral is made up of semicircles constructed as follows : the diameter [ A 0 A 1 ] of the semicircle C 0 has the middle A 2 ; the diameter [ A 1 A 2 ] of the semicircle C 1 has for middle A 3 ; In this way, we construct the semicircles C n ( n is a natural number) . The unit of length is the meter. We give A 0 A 1 =100 . 1 Note ‘ n the length of the semicircle C n . a Calculate ‘ 0 , ‘ 1 , ‘ 2 and ‘ 3 . b Express ‘ n +1 in terms of ‘ n . State the nature of the sequence ( ‘ n ) , specifying its reason. c Show that, for any integer n : ‘ n = 50 · ı · 1 2 n 2 The landscaper decides to draw only the eight semicircles C 0 , C 1 , C 2 , C 3 , . . . , C 7 . We call L the length of the spiral obtained with these eight semicircles. a Calculate: L = ‘ 0 + ‘ 1 + ‘ 2 + ··· + ‘ 7 . Give the rounding from L to 10 − 1 . b Knowing that the landscaper must plant a shrub ev-ery fifty centimetres from A 0 , deduce the number of shrubs to be planted. Formula : 1 + q + q 2 + · · · + q n = 1 − q n +1 1 − q ( q = 1) https://chingmath.fr chapExoCorrec/54 sacados/54 chapExoCorrec/30 sacados/30 C1C2C3C4A0A1A2A3
E.39 We divide an equilateral triangle into four equilateral triangles obtained by tracing the seg-ments joining the middles of the sides and blacken the central triangle. Each unblackened triangle is then divided into four equilateral triangles using the same procedure, and the central triangle is blackened as before. 1 a Perform the 3 ème step starting from an equilateral triangle with side 16 cm . How many blackened trian-gles have been added? b How many blackened triangles will be added in the fourth step? 2 Note T n the number of blackened triangles added in the n -th step where n is an integer greater than or equal to 1. The sequence ( T n ) thus defined, is a geometric sequence of reason 3. a Give the value of T 1 , T 2 , T 3 . b Express T n in terms of n . Check the results found in question 1 . 3 Calculate the total number of blackened triangles after the tenth step. E.7811 The minimum growth wage (SMIC) corresponds to the legal minimum hourly wage that employees must receive. Source : http://service-public.fr In 2018 , the net amount of the SMIC is 1187 e and the in-crease is set at 0 ; 9 % per year. Note u n the amount of the SMIC for the year 2018+ n . 1 What is the nature of the sequence u n ? Give the char-acteristic elements of the sequence u n . 2 a Give the expression of the terms of the sequence u n as a function of n . b Determine the amount of the minimum wage in 2030 , rounded to the nearest cent. 3 Using a calculator, determine the year in which the min-imum wage will exceed the amount of 1400 for the first time. e . 3. Homographic sequence E.2413 Consider the sequence u n n ∈ N is defined by the following recurrence relation: u n +1 = 4 u n +4 − u n +8 and whose first term is : u 0 = − 6 1 The graph below, represents the representative curve C f of the function f defined by: f ( x ) = 4 x + 4 − x + 8 a Place the first four terms of this sequence on the x-axis. b Make a conjecture as to the limit of the sequence when n tends to + ∞ . 2 Determine the value of the first three terms of the se-quence u n . 3 Consider the sequence v n ) n ∈ N whose rank term n is de-fined by: v n = 1 u n − 2 a Determine the value of the first three terms of the se-quence v n . b Establish the following relationship for any n ∈ N : v n +1 − v n = − 1 6 c Deduce the nature of the sequence v n and the ex-plicit formula defining the rank term n as a function of n . 4 a Establish the following formula : u n = 1+2 v n v n b Deduce the explicit formula defining the sequence u n . 5 Give the limit of the sequence u n . https://chingmath.fr chapExoCorrec/39 sacados/39 chapExoCorrec/7811 sacados/7811 chapExoCorrec/2413 sacados/2413 IJO
E.2416 Consider the sequence u n n ∈ N defined by the recurrence relation: u n +1 = 8 u n + 1 − u n + 10 whose first term has the value : u 0 = − 5 1 Give the value of the first three terms of the sequence u n . 2 Consider the sequence v n n ∈ N defined by the relation: v n = 1 u n − 1 a Determine the first three terms of the sequence v n . b Establish the following relationship : v n +1 − v n = − 1 9 c Give the nature of the sequence v n and give the ex-pression for the term v n as a function of n . 3 a Establish the following relationship : u n = 1 + v n v n b Express the term u n in terms of n . 4 Give the limit of the sequence u n when n tends to + ∞ . E.2983 In the plane provided with a reference frame O ; I ; J orthonormal, consider the curve C f represen-tative of the function f defined by the relation: f ( x ) = − 12 x + 36 x − 24 The line (Δ) is the first bisector of the plane. Consider the sequence u n n ∈ N defined by recurrence by: u n +1 = f u n ; u 0 = − 2 1 Graphically, place on the x-axis the first five values of the sequence u n . 2 Determine, by calculation, the value of the first three terms of the sequence u n . 3 We define the sequence w n n ∈ N defined by: w n = 1 u n − 6 a Establish the following equality: w n +1 − w n = − 1 18 b Give the nature and characteristics of the sequence w n as well as the expression of the term w n as a function of n . c Determine the expression of the term u n as a function of n . 4 Deduce the limit of the sequence u n 4. Adjoining suites E.654 Consider the two sequences ( u n ) and ( v n ) defined, for any natural number n , by: u 0 = 3 u n +1 = u n + v n 2 v 0 = 4 v n +1 = u n +1 + v n 2 1 Calculate u 1 , v 1 , u 2 and v 2 . 2 Let be the sequence ( w n ) defined for any natural number n by: w n = v n − u n . a Show that the sequence ( w n ) is a geometric sequence of reason 1 4 . b Express w n in terms of n and specify the limit of the sequence ( w n ) 3 After studying the direction of variation of the sequences ( u n ) and ( v n ) , demonstrate that these two sequences are adjacent. What can we deduce? https://chingmath.fr chapExoCorrec/2416 sacados/2416 chapExoCorrec/2983 sacados/2983 -2-123456I-1234567JOCf chapExoCorrec/654 sacados/654
E.3464 Consider the two sequences u n and v n defined for any natural number n by: u n = 1 n + 1 n + 1 + · · · + 1 2 n v n = 1 n + 1 + 1 n + 2 + · · · + 1 2 n 1 a Establish that the sequence u n is decreasing. b Establish that the sequence v n is increasing. 2 a Show that the sequences u n and v n are two ad-jacent sequences. b What can be said about the convergence of these two sequences? E.3465 Which of the following sequences defines two adjacent sequences : 1 u n = 3 n + 2 n + 1 et v n = 3 n + 7 n + 2 2 u n = 3 n + 4 n + 2 et v n = 4 n 2 + 1 n 2 + 1 E.2972 Consider the two sequences a n and b n defined by: a 0 = 1 ; b 0 = B ; a n +1 = 1 3 2 a n + b n b n +1 = 1 3 a n + 2 b n 1 Show that u n = b n − a n is a geometric sequence. 2 Show that : a n b n 3 direction of variation of a n and b n . 4 v n = a n + b n is constant? middle of [ A n B n ] 5 what is the convergence of ( a n ) and ( b n ) E.3466 Consider the function f defined on 0 ; + ∞ by: f ( x ) = 3 x − 2 x 1 a Draw up a table of variations for the function f . b Deduce that for any number x ∈ 3 2 ; 4 , we have : f ( x ) ∈ 3 2 ; 4 Consider the two sequences u n and v n defined by the relations : u 0 = 3 2 u n +1 = 3 · u n − 2 u n ; v 0 = 4 v n +1 = 3 · v n − 2 v n 2 Using recursive reasoning, establish that for any natural number n , we have : u n ∈ 3 2 ; 4 Similarly, we assume that for any n ∈ N : v n ∈ 3 2 ; 4 3 Determine the value of the first three terms of each of these sequences. 4 a Show that the sequence u n is increasing. b Show that the sequence v n is decreasing. c Establish the following framework, using recursive rea-soning, for any natural number n : 0 v n − u n 3 · 8 9 n d Establish that the two sequences u n and v n are adjacent. 5 a Justify that the limit ‘ verifies : ‘ = 3 · ‘ − 2 ‘ b Justify that the limit of the two sequences u n and v n is 2 . E.3467 Consider the two sequences u n and v n defined by: u 0 = − 3 u n +1 = 1 2 · u n + 2 ; v 0 = 5 v n +1 = 1 2 · v n + 2 1 Show that u n is increasing. 2 Show that v n is decreasing. 3 Show that, for any natural number, we have the frame : 0 v n − u n 8 · 1 2 n 4 Establish that the two suites u n are adjacent. 5 Deduce that these two sequences converge to the same number ‘ ; determine the value of ‘ . https://chingmath.fr chapExoCorrec/3464 sacados/3464 sacados/3465 sacados/2972 A refaire pique d'internet chapExoCorrec/3466 sacados/3466 chapExoCorrec/3467 sacados/3467
E.3479 Consider the sequences u n n ∈ N and v n n ∈ N defined by: u 0 = 2 ; ∀ n ∈ N ∗ ; v n = 2 u n and u n +1 = u n + v n 2 . 1 Calculate the first three terms of the sequences u n and v n . 2 Show that the sequences u n and v n are bounded above by 2 and bounded below by 1 . 3 Show that for all n ∈ N : u n +1 − v n +1 = ( u n − v n ) 2 2 · u n + v n 4 Show that for all n ∈ N : u n v n . 5 Show that u n is decreasing and v n is increasing. 6 Show that for all n ∈ N , u n − v n 1 and deduce that : u n − v n 2 u n − v n . 7 Show that for all n ∈ N : u n +1 − v n +1 1 4 u n − v n Deduce that for all n ∈ N : u n − v n 1 4 n 8 Show that the sequences u n and v n are adjacent and give their common limit ‘ . Does a convergent sequence of rational numbers have a rational number as its limit? E.3516 Consider the sequences u n n ∈ N and v n n ∈ N définies for any natural number n par : u 0 = 0 u n +1 = 3 · u n + 1 4 ∀ n ∈ N ; v 0 = 2 v n +1 = 3 · v n + 1 4 ∀ n ∈ N 1 Calculate u 1 , u 2 , u 3 d on the one hand and v 1 , v 2 , v 3 d on the other. 2 In an orthonormal frame of reference O ; −→ i ; −→ j (graphic unit : 5 cm ) trace the straight lines D and Δ d of respective equations : y = 3 · x + 1 4 ; y = x Use D and Δ to construct the points A 1 , A 2 , A 3 with respective abscissas u 1 , u 2 , u 3 as well as the points B 1 , B 2 , B 3 with respective abscissas v 1 , v 2 , v 3 . The plots will be drawn on graph paper . 3 We consider the sequence s n n ∈ N defined for all natural numbers n by: s n = u n + v n a Calculer s 0 , s 1 , s 2 , s 3 . Based on these results, what can we conjecture about the sequence s n ? b Using recursive reasoning, show that the sequence s n is a constant sequence. 4 Consider the sequence d n defined for all natural num-bers n by: d n = v n − u n a Show that the sequence d n is a geometric sequence. b Give the expression of d n in terms of n . 5 Using the results of questions 3 b and 4 b give the expression of u n and v n as a function of n . 6 Show that the sequences u n and v n convergent. Specify their limits. 5. Adjacent sequences - Annals E.3184 Part A Consider the sequences of points A n and B n defined for any natural number n as follows : on an axis oriented O ; −→ u given below, the point A 0 has abscissa 0 and the point B 0 has abscissa 12 . The point A n +1 is the barycenter of the points ( A n ; 2) and ( B n ; 1) , the point B n +1 is the barycenter of the weighted points ( A n ; 1) and ( B n ; 3) . 1 On the graph, place the points A 2 , B 2 . 2 We define the sequences a n and b n of the respective abscissas of the points A n and B n . Show that : a n +1 = 2 · a n + b n 3 We also admit that : b n +1 = a n +3 b n 4 Part B 1 Consider the sequence ( u n ) defined, for any natural num-ber n , by: u n = b n − a n a Show that the sequence ( u n ) is geometric. Specify the reason. b Give the expression of u n as a function of the natural number n . https://chingmath.fr chapExoCorrec/3479 sacados/3479 chapExoCorrec/3516 sacados/3516 Sportifs de haut niveau 1997 01212 chapExoCorrec/3184 sacados/3184 Antilles-Guyane Juin 2006 5 points A0A1B1B0−u024681012
c Determine the limit of u n . Interpret this result geo-metrically. 2 a Show that the sequence a n is increasing (the sign of u n ) may be used. b Study the variations of the sequence b n ) . 3 What can we deduce from the previous results about the convergence of sequences ( a n ) and ( b n ) ? Part C 1 Consider the sequence v n defined, for any natural num-ber n , by: v n = 3 a n + 4 b n Show that the sequence ( v n ) is constant. 2 Determine the limit of the sequences ( a n ) and ( b n ) . E.3227 Consider the two sequences ( u n ) and ( v n ) defined, for any natural number n , by: u 0 = 3 u n +1 = u n + v n 2 v 0 = 4 v n +1 = u n +1 + v n 2 1 Calculate u 1 , v 1 , u 2 and v 2 . 2 Let ( w n ) be the sequence defined for any natural number n by: w n = v n − u n . a Show that the sequence ( w n ) is a geometric sequence of reason 1 4 . b Express w n in terms of n and specify the limit of the sequence ( w n ) 3 After studying the direction of variation of the sequences ( u n ) and ( v n ) , demonstrate that these two sequences are adjacent. What can we deduce? 4 Now consider the sequence ( t n ) defined, for any natural number n , by: t n = u n + 2 v n 3 a Demonstrate that the sequence ( t n ) is constant. b Deduce the limit of the sequences ( u n ) and ( v n ) . E.3235 The graph in the appendix will be completed and submitted with the copy. Let the function f be defined on the interval [0 ; 2] by: f ( x ) = 2 x + 1 x + 1 1 Study the variations of f over the interval [0 ; 2] . Show that if x ∈ [1 ; 2] then f ( x ) ∈ [1 ; 2] . 2 ( u n ) and ( v n ) are two sequences defined on N by: u 0 =1 and for any natural number n : u n +1 = f ( u n ) . v 0 =2 and for any natural number n : v n +1 = f ( v n ) . a The graph given in the appendix represents the func-tion f over the interval [0 ; 2] . Construct the first three terms of each of the sequences ( u n ) and ( v n ) on the x-axis, leaving all construction lines visible. Based on this graph, what can be conjectured about the direction of variation and convergence of the se-quences ( u n ) and ( v n ) ? b Show using recursive reasoning that : For any natural number n : 1 v n 2 . For any natural number n : v n +1 v n . We will assume that we can demonstrate in the same way that : For any natural number n : 1 u n 2 . For any natural number n : u n u n +1 . c Show that for any natural number n : v n +1 − u n +1 = v n − u n ( v n + 1)( u n + 1) . Deduce that for any natural number n : v n − u n 0 ; v n +1 − u n +1 1 4 ( v n − u n ) d Show that for all n ∈ N : v n − u n 1 4 n . e Show that the sequences ( u n ) and ( v n ) converge to the same real number ¸ . Determine the exact value of ¸ . https://chingmath.fr chapExoCorrec/3227 sacados/3227 chapExoCorrec/3235 sacados/3235 0.51.52I0.51.52JO
E.3270 We define the sequences ( a n ) and ( b n ) by a 0 =1 , b 0 =7 and a n +1 = 1 3 · 2 · a n + b n b n +1 = 1 3 · a n + 2 · b n for all n ∈ N Let D be a line equipped with a coordinate system O ; −→ i . For all n ∈ N , we consider the points A n and B n of respective abscissas a n and b n . 1 Place points A 0 , B 0 , A 1 , B 1 , A 2 and B 2 . 2 Let ( u n ) be the sequence defined by u n = b n − a n for any n ∈ N . Show that ( u n ) is a geometric sequence whose reason and first term will be specified. Express u n as a function of n . 3 Compare a n and b n . Study the direction of variation of the sequences ( a n ) and ( b n ) . Interpret these results geometrically. 4 Demonstrate that the sequences ( a n ) and ( b n ) are adja-cent. 5 Let ( v n ) be the sequence defined by v n = a n + b n for all n ∈ N . Show that ( v n ) is a constant sequence. Deduce that the segments A n B n all have the same middle I . 6 Prove that the sequences ( a n ) and ( b n ) are convergent and calculate their limit. Interpret this result geometri-cally. E.3243 This exercise is an organized re-view of knowledge. Part A : Course question The following results are assumed to be known : (1) two sequences ( u n ) and ( v n ) are adjacent when : one is increasing, the other is decreasing, and u n − v n tends to-wards 0 when n tends towards + ∞ ; (2) if ( u n ) and ( v n ) are two adjacent sequences such that ( u n ) is increasing and ( v n ) is decreasing, then for any n belonging to N , we have : u n v n ; (3) every increasing and bounded sequence is convergent ; ev-ery decreasing and bounded sequence is convergent. Prove the following proposition : ˇ Two adjacent sequences are convergent and have the same limit ı. Part B Consider a sequence ( u n ) , defined on N , none of whose terms is zero. We then define the sequence ( v n ) on N by v n = − 2 u n . For each statement, indicate whether it is true or false and provide a proof for the answer given. In the case of a false statement, the proof will consist of providing a counterexam-ple. An answer without proof will not earn any points. 1 If ( u n ) is convergent, then ( v n ) is convergent. 2 If ( u n ) is bounded below by 2, then ( v n ) is bounded below by − 1 . 3 If ( u n ) is decreasing, then ( v n ) is increasing. 4 If ( u n ) diverges, then ( v n ) converges to zero. https://chingmath.fr chapExoCorrec/3270 sacados/3270 Antilles-Guyane Juin 2004 4 points chapExoCorrec/3243 sacados/3243 France Juin 2005 4 points
E.3997 1 The sequence u is defined by: u 0 = 2 ; u n +1 = 1 3 · u n + 23 27 for all n ∈ N . a We have represented in a direct orthonormal coor-dinate system of the plane, the line with equation y = 1 3 · x + 23 27 and the point A with coordinates (2 ; 0) . Construct the first four terms of the sequence u on the x-axis. b Show that if the sequence u converges, then its limit is ‘ = 23 18 . c Prove that for any natural number n , we have : u n 23 18 d Study the monotonicity of the sequence u and give its limit. 2 a So n a natural number greater than or equal to 1 . Démontrer que : n +1 k =2 1 10 k = 1 90 · 1 − 1 10 n . c’est-à-dire que : 1 10 2 + 1 10 3 + · · · + 1 10 n +1 = 1 90 · 1 − 1 10 n b The sequence v is defined by v n =1.2777 ::: 7 avec n déci-males consecutive equal to 7 . Thus : v 0 = 1.2 ; v 1 .27 ; v 2 = 1.277 Using question a demonstrate that the limit of the sequence v is a rational number r (i.e. the quotient of two integers.) . 3 Is the sequence u defined at 1 and the sequence v sont adjacent? Justify. E.3245 The four questions in this exercise are independent and are each worth one point. For each question, there are exactly two correct statements. Candidates must indicate the two correct statements on their answer sheet. No justification is required. Each correct answer is worth 0.5 points, and each incorrect answer deducts 0.25 points. Giving three or more statements for a question, or giving none at all, earns no points. If, when this scoring system is applied, the total points for the exercise are negative, they are reduced to zero. 1 The following sequences are convergent : a 2 n n 2005 n> 0 b 2 n + ( − 1) n · √ n n + 1 n ∈ N c n · sin 1 n n> 0 d n ln n n> 1 2 We construct three sequences ( u n ) , ( v n ) and ( w n ) having, for any natural number n , the following properties : u n v n w n lim n ↦→ + ∞ ( u n ) = − 1 ; lim n ↦→ + ∞ ( w n ) = 1 Then : a lim n ↦→ + ∞ ( v n ) = 0 b The sequence ( u n ) is bounded below. c For all n ∈ N , we have : − 1 v n 1 . d It is unclear whether the sequence ( v n ) has a limit or not. 3 A sequence ( u n ) is defined on N by: u 0 = 1.5 u n +1 = 2 u n − 1 for all n ∈ N a The sequence ( u n ) converges to 1 , the abscissa of the intersection point of the lines with equations y = x and y =2 x − 1 . b The sequence ( v n ) , defined on N by v n = u n − 1 , is geo-metric. c The sequence ( v n ) is bounded above. d The sequence ( w n ) , defined on N by w n =ln( u n − 1) , is arithmetic. 4 Two sequences ( x n ) and ( y n ) are defined for n> 0 by the relations : x n = 1 n + 1 n + 1 + · · · + 1 2 n y n = 1 n + 1 + 1 n + 2 + · · · + 1 2 n a The sequences ( x n ) and ( y n ) are both increasing. b x 3 = 19 20 and y 3 = 37 60 . c The sequences ( x n ) and ( y n ) are not bounded above. d The sequences ( x n ) and ( y n ) are adjacent. https://chingmath.fr chapExoCorrec/3997 sacados/3997 23I2JOij chapExoCorrec/3245 sacados/3245
E.3237 1 Demonstrate that for any n of N ∗ and any x of [0 ; 1] : 1 n − x n 2 1 x + n 1 n 2 a Calculate 1 0 1 x + n dx b Deduce using 1 , that for any n ∈ N ∗ : 1 n − 1 2 n 2 ln n + 1 n since ln n + 1 n 1 n 3 We call U the sequence defined for n ∈ N ∗ by: U n = k = n k =1 1 k − ln( n ) = 1 + 1 2 + 1 3 + · · · + 1 n − ln( n ) Show that U is decreasing (we can use 2 b ) 4 We denote by V the sequence of general term : V ( n ) = k = n k =1 1 k − ln( n +1) = 1+ 1 2 + 1 3 + · · · + 1 n − ln( n +1) Show that V is increasing. 5 Show that U and V converge to a common limit denoted ‚ . By the method of your choice, determine the value of ‚ rounded to the nearest 10 − 2 . 6. Continuation: crossing the limit E.3413 The sequence u is defined by: u 0 = 2 u n +1 = 1 3 · u n + 23 27 pour any natural number n 1 We have represented in a direct orthonormal reference frame of the plane in the appendix, the line with equa-tion y = 1 3 x + 23 27 and the point A with coordinates (2 ; 0) . Construct on the x-axis the first four terms of the se-quence u . 2 Show that if the sequence u is convergent then its limit is ‘ = 23 18 . 3 Demonstrate that for any natural number n , we have : u n 23 18 . 4 Study the monotonicity of the sequence u and give its limit. E.3426 Consider the sequence u n n ∈ N defined by: u 0 = 4 u n +1 = 2 · u n − 1 for all n ∈ N The aim of the exercise is to show that the sequence u n is divergent ; to do this, let us reason by the absurd, suppose that the sequence u n converges to ‘ : 1 Determine the value of ‘ . Consider the sequence v n n ∈ N defined by: v n = u n − ‘ for all n ∈ N 2 Establish that the sequence v n is a geometric sequence whose characteristic elements will be specified. 3 Deduce that the sequence u n is divergent. E.3427 Consider the sequence u n n ∈ N defined by: u 0 = 7 u n +1 = 2 3 · u n − 1 for all n ∈ N 1 In this question, it is assumed that the sequence u n converges. Let ‘ be its convergence value. Determine, by a passage to the limit, the value of ‘ . 2 Consider the sequence v n defined for any natural num-ber n by: v n = u n +3 a Show that the sequence ( v n ) is a geometric sequence whose characteristics will be specified. b Deduce the expression of the term u n as a function of n . c Establish the convergence of the sequence u n . https://chingmath.fr chapExoCorrec/3237 sacados/3237 Antilles - Guyane Juin 2005 6 points chapExoCorrec/3413 sacados/3413 23I2JO chapExoCorrec/3426 sacados/3426 chapExoCorrec/3427 sacados/3427
E.3456 1 If a sequence u n converges to ‘ , to what value does the expression 1 3 · u n + 1 3 converge? 2 Consider the sequence u n n ∈ N defined by: u 0 = 5 u n +1 = 1 3 · u n + 1 3 for all n ∈ N We assume that the sequence u n converges to a number denoted by ‘ . a Determine the exact value of the first eight terms of the sequence u n . b Using a calculator, complete the following table, enter-ing values to the nearest 10 − 2 . n u n 1 3 · u n + 1 3 0 1 2 3 4 5 6 7 c Make a conjecture about the following two limits: lim n ↦→ + ∞ u n ; lim n ↦→ + ∞ 1 3 · u n + 1 3 d Solve the equation : x = 1 3 · x + 1 3 e Deduce the value of the limit ‘ . E.3471 The curve C representing the function f and the line Δ are plotted on the graph below : Consider the function g defined on 0 ; + ∞ par : g ( x ) = f ( x ) − x The following properties are assumed : The function f is strictly increasing on 0 ; + ∞ . The function g is strictly decreasing on 0 ; + ∞ . There exists a unique real number ¸ such that g ( ¸ )=0 ; the number ¸ belongs to the interval [2 ; 3] . Consider the sequence u n defined by u 0 =1 and for any nat-ural number n by: u n +1 = f u n 1 Starting from u 0 , using the curve C and the line Δ , we placed u 1 on the x-axis. In the same way, place the terms u 2 and u 3 on the x-axis, leaving the construction lines visible. 2 Place the point I on the curve C whose x-coordinate is ¸ . 3 a Show that, for any natural number n , we have : 1 u n ¸ b Prove that the sequence u n converges. c Determine its limit. 7. Sequence and limit crossing - Annales E.3148 Let the sequence u n be defined for any natural number n by: u 0 = 1 2 et u n +1 = 1 2 u n + 2 u n 1 a Let f be the function defined on 0 ; + ∞ by: f ( x ) = 1 2 x + 2 x Study the direction of variation of f , and plot its repre-sentative curve in the plane provided with an orthonor-mal reference frame O ; −→ i ; −→ j . (We’ll take 2 cm as the unit) . b Use the previous graph to construct the points A 0 , A 1 , A 2 and A 3 on the axis O ; −→ i of respective abscissas u 0 , u 1 , u 2 and u 3 . 2 a Show that for any n ∈ N ∗ : u n 2 . b Show that for any x 2 : f ( x ) x . c Deduce that the sequence ( u n ) is decreasing from rank 1. d Prove that it converges. 3 Let ‘ be the limit of the sequence u n . Show that ‘ is a solution of the equation : x = 1 2 x + 2 x https://chingmath.fr chapExoCorrec/3456 sacados/3456 chapExoCorrec/3471 sacados/3471 ijCu0u1 chapExoCorrec/3148 sacados/3148
Deduce its value. E.3902 Let ( u n ) be the sequence defined by u 0 =5 and for any integer n by: u n +1 = 4 · u n − 1 u n + 2 If f is the function defined on the interval − 2 ; + ∞ by f ( x )= 4 x − 1 x +2 then we have, for any natural number n : u n +1 = f ( u n ) . Part of the representative curve C of the function f is given in the appendix, along with the line Δ with equation y = x . 1 a On the x-axis, place u 0 then construct u 1 , u 2 and u 3 leaving the construction lines visible. b What conjectures can be made about the direction of variation and convergence of the sequence ( u n ) ? 2 a Prove by recurrence that, for any natural number n , we have : u n − 1 > 0 b In this question, any evidence of research, even if in-complete, or initiative, even if unsuccessful, will be taken into account in the assessment. Validate the conjectures made in question 1 b 3 In this question, we propose to study the sequence u n using another method, by determining an expression for u n in terms of n . For any natural number n , we set v n = 1 u n − 1 a Prove that the sequence v n is an arithmetic sequence with common difference 1 3 . b For any natural number n , express v n then u n in terms of n . c Deduce the limit of the sequence u n . E.5017 Let f be the function defined on the interval − 1 ; + ∞ par : f ( x ) = 3 − 4 x + 1 Consider the sequence defined for all n ∈ N par : u 0 = 4 ; u n +1 = f ( u n ) pour all n ∈ N 1 We have drawn, below, the curve C représentative of the function f sur the interval 0 ; + ∞ and the straight line ( d ) d of equation y = x . a On the graph, place on the x-axis u 0 , u 1 , u 2 et u 3 . Show the construction lines. b What can we conjecture about the direction of varia-tion and convergence of the sequence u n ? 2 In this question, we will demonstrate the conjectures for-mulated in question 1 b : a Demonstrate by recursive reasoning that u n 1 for all n ∈ N . b Show that the function f is increasing on 0 ; + ∞ . De-duce that for any natural number n , we have : u n +1 u n . c Deduce from the previous questions that the sequence u n is convergent and calculate its limit. https://chingmath.fr chapExoCorrec/3902 sacados/3902 -2-10123456-2-11234C chapExoCorrec/5017 sacados/5017 234I23JOC(d
E.3242 I. Part One We call f and g the two functions defined on the interval 0 ; + ∞ by: f ( x ) = ln(1+ x ) − x ; g ( x ) = ln(1+ x ) − x + x 2 2 1 Study the variations of f and g on 0 ; + ∞ . 2 Deduce that for all x 0 : x − x 2 2 ln(1+ x ) x II. Part Two We propose to study the sequence ( u n ) of real numbers de-fined by: u 1 = 3 2 ; u n +1 = u n 1 + 1 2 n +1 1 Show by recurrence that u n > 0 for all natural numbers n 1 . 2 Show by recurrence that for any natural number n 1 : ln u n = ln 1+ 1 2 + ln 1+ 1 2 2 + · · · + ln 1+ 1 2 n 3 We set : S n = 1 2 + 1 2 2 + 1 2 3 + · · · + 1 2 n T n = 1 4 + 1 4 2 + 1 4 3 + · · · + 1 4 n Using the first part, show that : S n − 1 2 T n ln u n S n 4 Calculate S n and T n based on n . Deduce lim n ↦→ + ∞ S n and lim n ↦→ + ∞ T n . 5 Study of the convergence of the sequence ( u n ) . a Show that the sequence ( u n ) is strictly increasing. b Deduce that ( u n ) is convergent. Let ‘ be its limit. c We accept the following result : if two sequences ( v n ) and ( w n ) are convergent and such that v n w n for all n natural numbers, then : lim n ↦→ + ∞ v n lim n ↦→ + ∞ w n Show that 5 6 ln ‘ 1 and deduce a framework of ‘ . 8. Former annuals (before 2012) E.5142 Part A : study of a function Consider the function f defined on 1 ; + ∞ par : f ( x ) = x ln x On the attached appendix, the curve C representative of the function f as well as the straight line D of equation y = x have been plotted in an orthogonal reference frame. 1 Calculate the limits of the function f en + ∞ et in 1 . 2 Study the variations of the function f on the interval 1 ; + ∞ . 3 Determine that if x e then f ( x ) e . Part B: study of a recurrent sequence Consider the sequence u n defined by: u 0 = 5 ; u n +1 = f u n pour tout n ∈ N 1 On the attached appendix, to be returned with the copy, using the curve C and the line D , place the points A 0 , A 1 and A 2 with zero ordinates and respective abscissas u 0 , u 1 and u 2 . The construction lines will be left visible. What conjectures can be made about the variations and convergence of the sequence u n ? 2 a Show that, for any natural number n , we have : u n e . b Determine the variations of the sequence u n . c Deduce that the sequence u n is convergent. d Determine its limit ‘ . 3 The following algorithm is given : X ← 5 Y ← 0 As long as X>2.72 X ← X ln X Y ← Y+1 End As long as Using the following table, obtained with a spreadsheet, determine the value of the variable Y at the end of the execution of this algorithm: n u n 0 5 1 3 ; 106 674 672 8 2 2.740 652 532 3 3 2.718 372 634 6 4 2.718 281 830 01 5 2.718 281 828 https://chingmath.fr chapExoCorrec/3242 sacados/3242 chapExoCorrec/5142 sacados/5142 Antilles - Guyanne Septembre 2012
E.3964 Consider the equation : ( E ): ln x = − x The aim of the exercise is to prove that the equation ( E ) has a unique solution denoted by ¸ belonging to the interval 0 ; + ∞ and to use a convergent sequence to obtain a bound. Part A: existence and uniqueness of the solution Consider the function f defined on ]0 ; + ∞ by: f ( x ) = x + ln x . 1 Determine the direction of variation of the function f over the interval 0 ; + ∞ . 2 Show that the equation f ( x )=0 has a unique solution denoted by ¸ belonging to the interval 0 ; + ∞ . 3 Check that : 1 2 ¸ 1 Part B: framing the solution ¸ Consider the function g defined on 0 ; + ∞ by: g ( x ) = 4 · x − ln x 5 1 Study of some properties of the function g . a Study the direction of variation of the function g on the interval 0 ; + ∞ . b Deduce that for any real number x belonging to the interval 1 2 ; 1 , g ( x ) belongs to this interval. c Prove that a real number x belonging to the interval 0 ; + ∞ is a solution to the equation ( E ) if, and only if, g ( x )= x . 2 Consider the sequence u n defined by u 0 = 1 2 and for any natural number n , by: u n +1 = g ( u n ) a Using the direction of variation of the function g , prove by induction that for any natural number n : 1 2 u n u n +1 1 b Deduce that the sequence u n converges to ¸ . 3 Search for an approximate value of ¸ : a Using the calculator, determine the value u 10 , rounded to the sixth decimal place. b We assume that u 10 is an approximate value by default at 5 × 10 − 4 close to ¸ . Deduce a bound for ¸ in the form : u ¸ v where u and v are two decimals written with three decimal places. E.4081 Let u n be the sequence defined by: u 0 = 1 u n +1 = u n − ln u 2 n +1 for any natural number n . Part A Let f be the function defined on R by: f ( x )= x − ln x 2 +1 1 Solve in R the equation f ( x )= x . 2 Study the direction of variation of the function f on the interval 0 ; 1 . Deduce that if x ∈ 0 ; 1 then f ( x ) ∈ 0 ; 1 Part B 1 Show by recurrence that, for any integer n 0 : u n ∈ 0 ; 1 2 Study the direction of variation of the sequence u n . 3 Demonstrate that the sequence u n is convergent. De-termine its limit. https://chingmath.fr 2345678I2345678JOCD chapExoCorrec/3964 sacados/3964 Asie Juin 2009 6 points chapExoCorrec/4081 sacados/4081 Nouvelle-Caledonie Mars 2011 5 points
E.5152 Part A Consider the function f defined and derivable on the interval 0 ; + ∞ par : f ( x ) = 5 · ln( x +3) − x 1 a On calls f la derivative function of the function f sur 0 ; + ∞ . Calculate f ( x ) and study its sign on 0 ; + ∞ . b Give, in a table, the variations of f on the interval 0 ; + ∞ . c Show that, for any x strictly positive, we have : f ( x ) = x · 5 · ln x x − 1 + 5 · ln 1+ 3 x d Infer the limit of f en + ∞ . e Complete the table of variations of f over the interval 0 ; + ∞ . 2 a Montrer que l’équation f ( x )=0 admet une unique solution dans l’intervalle 0 ; + ∞ . We will note ¸ this solution. b After checking that ¸ belongs to the interval 14 ; 15 , give the value of ¸ arrondie to 10 − 2 près. c Determine the sign of f over the interval 0 ; + ∞ . Part B Let u n be the sequence defined by: u 0 = 4 ; u n +1 =5 · ln u n +3 for all n ∈ N ∗ Consider the function g defined on the interval 0 ; + ∞ par : g ( x ) = 5 ln( x +3) Below, we have plotted in an orthonormal coordinate system the line D with equation y = x and the curve C , which repre-sents the function g . 1 a Construct the terms u 0 , u 1 , u 2 from the sequence u n using the given line and curve and leaving the construction lines visible. b Formulate a conjecture about the direction of variation of the sequence u n . 2 a Study the meaning of variations in the function g over the interval 0 ; + ∞ . b Verify that g ( ¸ )= ¸ where ¸ is defined in part A ques-tion 2 a . c Prove by recurrence that, for any natural number n , we have : 0 u n a d Then prove the conjecture made in question 1 b of part B . e Using question 2 a from part A , justify that : lim n ↦→ + ∞ u n = ¸ 3 Consider the following algorithm u ← 4 As long as u − 14.2<0 u ← 5 · ln(u+3) End As long as a In this question, any attempt at research, even if in-complete or unsuccessful, will be taken into account in the assessment. Justify that the execution of this algorithm terminates. b Give the value of the variable u at the end of the exe-cution of this algorithm (rounded to 5 decimal places) . E.3200 Part A Let f be the function defined on the interval 1 ; + ∞ by: f ( x ) = x ln x 1 a Determine the limits of the function f at 1 and at + ∞ . b Study the variations of the function f . 2 Let ( u n ) be the sequence defined by: u 0 = 5 ; u n +1 = f ( u n ) for all n ∈ N . a We have plotted the representative curve C of the func-tion f in the figure given in the appendix, which will be returned with the copy. Construct the line with equation y = x and the points M 1 and M 2 on the curve C with respective abscissas u 1 and u 2 . Propose a con-jecture about the behavior of the sequence ( u n ) . b Prove that for any natural number n , we have u n e (you may use question 1 b ) c Prove that the sequence u n converges to a real num-ber ‘ in the interval e ; + ∞ . Part B Recall that the function f is continuous on the interval 1 ; + ∞ : 1 By studying the limit of the sequence f ( u n ) in two ways, demonstrate that : f ( ‘ )= ‘ . 2 Deduce the value of ‘ . https://chingmath.fr chapExoCorrec/5152 sacados/5152 0246810121416182022246810121416CD chapExoCorrec/3200 sacados/3200
E.3228 Let f be the function defined on [0 ; + ∞ [ by: f ( x ) = x · e − x Let Γ be the representative curve of the function f in an orthonormal coordinate system O ; −→ i ; −→ j (graphical unit 10 cm ) . Part A. 1 a Determine the limit of f at + ∞ . b Study the variations of f and draw up its variation table. c Construct Γ in the coordinate system O ; −→ i ; −→ j . 2 a Show that, for any real number m in 0 ; 1 e , the equation f ( x )= m has two solutions. b In the case where m = 1 4 , we call the solutions (with ¸<˛ ) ¸ and ˛ . Determine an interval of amplitude 10 − 2 for ¸ . c Solve the equation f ( x )= m in the case where m =0 and m = 1 e . Part B. 1 Consider the sequence ( u n ) defined on N by: u 0 = ¸ u n +1 = u n e − u n for all natural numbers n where ¸ is the real number defined in question A 2 b a Show by recurrence that, for all n ∈ N : u n > 0 . b Show that the sequence ( u n ) is decreasing. c Is the sequence ( u n ) convergent? If so, determine its limit. 2 Consider the sequence ( w n ) defined on N by: w n =ln u n . a Show that, for all natural numbers n , we have : u n = w n − w n +1 b We set : S n = u 0 + u 1 + ··· + u n . Show that : S n = w 0 − w n +1 . c Deduce : lim n ↦→ + ∞ S n . 3 Consider the sequence ( v n ) defined on N by its first term v 0 ( v 0 > 0 ) and, for any natural number n : v n +1 = v n · e − v n Is there a value of v 0 other than ¸ such that, for all n 1 , we have : u n = v n ? If so, specify which one. 9. Homographic and geometric sequences E.5734 Consider the sequence u n de-fined by: u 0 = 2 ; u n +1 = 1 + 0.5 · u n 0.5 + u n for all n ∈ N We assume that for any natural number n , we have : u n > 1 1 Consider the sequence v n defined, for any natural num-ber n , by: v n = u n − 1 u n + 1 a Prove that the sequence v n is geometric with com-mon ratio − 1 3 . b Calculate v 0 then write v n as a function of n . 2 a Show that, for any natural number n we have : v n = 1 . b Show that, for any natural number n we have : u n = 1 + v n 1 − v n . c Determine the limit of the sequence u n . E.5739 Consider the sequence u n de-fined by: u 0 = 1 2 ; u n +1 = 3 · u n 1 + 2 · u n for any n ∈ N We admit that for any natural number n , we have : 0 < u n < 1 Let v n be the sequence defined, for any natural number n , by: v n = u n 1 − u n 1 Show that the sequence v n is a geometric sequence of reason 3 . 2 Express for any natural number n , v n as a function of n . 3 Deduce that, for any natural number n : u n = 3 n 3 n + 1 4 Determine the limit of the sequence u n . https://chingmath.fr 2345I2345JOC chapExoCorrec/3228 sacados/3228 chapExoCorrec/5734 sacados/5734 Extrait d'Asie Juin 2013 chapExoCorrec/5739 sacados/5739
E.6737 Consider the sequence u n defined on N by the relation: u 0 = 3 ; u n +1 = 2 − u n u n for everything n ∈ N We admit that the terms of the suite u n are different from − 2 and from 0 . 1 We define the sequence v n defined by the relation: v n = u n − 1 u n + 2 for all n ∈ N a Justify that the sequence v n is a geometric sequence whose characteristic elements will be specified. b Express the terms of the sequence v n as a function of n . 2 Deduce an expression for the terms of the sequence u n as a function of n . 3 Determine the limit of the terms of the sequence u n when n tends to + ∞ . E.2415 Consider the sequence u n n ∈ N defined by: u 0 = 13 2 ; u n +1 = 10 · u n + 1 u n + 10 for any n ∈ N We admit that the terms of the sequence u n are strictly greater than 1 . 1 Consider the function f defined by f ( x )= 10 · x +1 x +10 Below, in the orthonormal frame ( O ; I ; J ) , the curve C f representative of the function f and the first bisector (Δ) of the plane. a Show the first five terms of the sequence u n on the x-axis b Make a conjecture as to the limit of the sequence u n when n tends to + ∞ . 2 Consider the sequence v n n ∈ N defined by the relation: v n = u n − 1 u n + 1 for any integer n ∈ N . a Show that the sequence v n is a geometric sequence. b Give the expression of the terms of the sequence v n as a function of their rank n . 3 a Establish the following equality for any natural num-ber n : u n = 1 + v n 1 − v n b Deduce the expression of the terms of the sequence u n as a function of their rank n . 4 Determine the limit of u n when n tends to + ∞ . E.2984 In the plane provided with a O ; I ; J orthonormal, consider the curve C f of the function f whose image of a number x is defined by: f ( x ) = − x + 4 x + 2 The line (Δ) is the first bisector of the plane. Consider the sequence u n n ∈ N defined by the relation: u 0 = 8 ; u n +1 = f u n for any natural number n . We admit that the terms of the sequence u n are defined and that for any natural number n , we have : u n > − 1 1 On the x-axis, show the values of the first six terms of the sequence. 2 Determine, by calculation, the first three terms of the sequence u n . 3 Consider the sequence w n n ∈ N defined by the relation: w n = u n − 1 u n + 4 a Establish the following equality: w n +1 = − 2 u n +2 3 u n +12 b Deduce that the sequence w n is a geometric sequence whose characteristics will be specified. c Give the expression of the term w n as a function of n . 4 We admit that for any n ∈ N , we have : − 1 <w n < 1 a Establish equality: u n = − 4 − 5 w n − 1 b Deduce the explicit formula for the sequence u n 5 Deduce the limit of the sequence w n . https://chingmath.fr chapExoCorrec/6737 sacados/6737 chapExoCorrec/2415 sacados/2415 IJOCf chapExoCorrec/2984 sacados/2984 -12345678I-1234JOCf
E.3517 A numerical sequence U n n ∈ N ∗ is defined by its first term U 1 and the recurrence relation: U n +1 = 6 + U n 2 + U n pour tout n ∈ N 1 Show that there are two values a and b of U 1 ( a<b ) for which the sequence is constant. 2 a Show that if U 1 = a and U 1 = b , the same is true for U n . b Under these conditions, calculate: U n +1 − a U n +1 − b depending on U n − a U n − b . 3 We assume that U 1 = a and U 1 = b and we consider the sequence V n n ∈ N ∗ defined by: V n = U n − a U n − b for all natural numbers n a Show that the sequence V n is a geometric sequence. b Calculate the limit of ⏐ ⏐ V n ⏐ ⏐ when n tends to infinity. Deduce that of U n . 10. Homographic and arithmetic sequences E.5852 Let u n be the sequence defined by: u 0 =5 ; u n +1 = 4 · u n − 1 u n + 2 for any natural number n . We admit that for any natural number n , we have : u n > 1 We define the terms of the sequence u n by the relation: v n = 1 u n − 1 for any natural number n . 1 Demonstrate that the sequence v n is an arithmetic se-quence of reason 1 3 . 2 For any natural number ; , express v n , then u n as a func-tion of n . 3 Deduce the limit of the sequence u n E.6736 Consider the sequence u n defined on N by the relation: u 0 = 4 ; u n +1 = − u n − 2 2 · u n + 3 for everything n ∈ N We admit that for all terms of the sequence u n are defined and that they verify: u n > − 1 1 We define the sequence v n defined by the relation: v n = 1 u n + 1 for all n ∈ N a Justify that the sequence v n is an arithmetic se-quence whose characteristic elements will be deter-mined. b Express the terms of the sequence v n as a function of n . 2 a Establish equality: u n = 4 − 10 · n 1 + 10 · n for all n ∈ N b Show that the terms of the sequence u n , for n 1 , admit the expression : u n = 4 n − 10 1 n + 10 c Determine the limit of the terms of the sequence u n when n tends to + ∞ . E.6735 Consider the sequence u n de-fined on N by the relation: u 0 = 5 ; u n +1 = 3 · u n − 1 u n + 1 for any n ∈ N We admit that the sequence u n is defined for any integer n ∈ N and its terms verify: u n > 1 1 We define the sequence v n defined by the relation: v n = 1 u n − 1 for all n ∈ N a Justify that the sequence v n is an arithmetic se-quence of reason 1 2 . Specify the first term of this se-quence. b Express the terms of the sequence v n as a function of n . 2 a Deduce the following expression for the terms in the sequence u n : u n = 5+2 · n 1+2 · n b Establish, for any n ∈ N ∗ , the relation: u n = 2+ 5 n 2+ 1 n c Determine the limit of the terms of the sequence u n when n tends to + ∞ . https://chingmath.fr chapExoCorrec/3517 sacados/3517 Rouen 1976 chapExoCorrec/5852 sacados/5852 chapExoCorrec/6736 sacados/6736 chapExoCorrec/6735 sacados/6735 fichierPlus/6735/
E.3400 In the plane provided with an orthonor-mal O ;I ; J reference frame, consider the C f curve repre-sentative of the f function defined by the relation: f ( x ) = − 12 x + 36 x − 24 The line (Δ) is the first bisector of the plane. Consider the sequence u n n ∈ N defined by recurrence by: u 0 = − 2 ; u n +1 = f u n for all n ∈ N . We admit that the sequence u n is defined on N and that its terms verify the comparison : u n < 6 1 Graphically, place the first five values of the sequence u n on the x-axis. 2 Calculate the values of the first three terms of the se- quence u n . 3 We define the sequence w n n ∈ N defined by: w n = 1 u n − 6 for all n ∈ N a Establish the following equality: w n +1 − w n = − 1 18 b Give the nature and characteristics of the sequence w n as well as the expression of the term w n in terms of n . c Determine the expression of the term u n in terms of n . 4 Deduce the limit of the sequence u n E.5735 Consider the numerical sequence v n defined by: v 0 = 1 ; v n +1 = 9 6 − v n for any natural number n . 1 a Prove by recurrence that, for any natural number n : 0 <v n < 3 b Demonstrate that, for any natural number n : v n +1 − v n = (3 − v n ) 2 6 − v n c Demonstrate that the sequence v n is convergent. 2 Consider the sequence w n defined for any natural inte-ger n by: w n = 1 v n − 3 a Demonstrate that w n is an arithmetic sequence of reason − 1 3 . b Deduce the expression of w n , then that of v n as a function of n . c Determine the limit of the sequence v n . https://chingmath.fr chapExoCorrec/3400 sacados/3400 -2-123456I-1234567JOCf chapExoCorrec/5735 sacados/5735 Extrait Liban Mai 2013