Outside the high school program / Arithmetic and geometric sequences 56 exercises (including 45 corrected)

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1. Arithmetic sequences: first terms E.4582 Consider the sequence u n , where n ∈ N , defined by the recurrence relation: u 0 = 5 ; u n +1 = u n − 2 1 Give the nature of the sequence u n and its characteris- tic elements. 2 Give the explicit formula giving the value of u n as a func-tion of n . 3 Determine the value of u 20 . 2. Arithmetic sequences: characteristic features E.57 Let ( u n ) be a sequence verifying for any natural number n : u n +1 − u n = r where r ∈ R Show by recurrence that for any n ∈ N , we have : u n +1 = u 0 + n × r E.7593 Consider the sequence u n defined for any positive integer or zero ( n ∈ N ) arithmetic whose value of the following two terms is known : u 7 =3 ; u 22 =15 Determine the characteristic elements of the terms in the se-quence u n . E.7594 Consider the sequence u n defined for any positive integer or zero ( n ∈ N ) arithmetic whose value of the following two terms is known : u 9 =6 ; u 24 =8.1 Determine the characteristic elements of the terms in the se-quence u n . E.4577 Consider the sequence u n arithmetic, defined for any n ∈ N , for which we have the values of the two terms : u 4 = 3 ; u 7 = 15 1 Determine the characteristic elements of the suite u n . 2 Give the recurrence formula, then the explicit formula characterizing the sequence u n E.4618 Consider the sequence u n arithmetic, defined for any n ∈ N , for which the values of the two terms are known : u 7 = 23 ; u 13 = − 1 1 Determine the characteristic elements of the suite u n . Justify your approach. 2 Give the explicit formula defining a term in the sequence u n as a function of its rank n . E.9494 Consider the arithmetic sequence v n defined by: v 0 = 6 ; v n +1 = v n − 2 for all n ∈ N 1 Give the characteristic elements of the sequence v n . 2 Determine the value of the 6 first terms of the sequence v n . 3. Arithmetic sequences: recognizing E.4574 Consider the sequence u n , defined for n ∈ N , whose first terms are: u 0 = 2 ; u 1 = 5 ; u 2 = 9 ; u 3 = 12 Justify that the sequence u n is not an arithmetic sequence. E.7596 Consider the sequence u n , defined for n ∈ N , whose first terms are: u 0 = 5 ; u 1 = 6.5 ; u 2 = 7 ; u 3 = 8.5 Justify that the sequence u n is not an arithmetic sequence. 4. Geometric sequences: first terms E.4573 Consider the sequence u n geometric, defined for any n ∈ N , of first term 2 and reason 3 . Determine the first five terms of this sequence. 5. Geometric sequences: characteristic features E.48 Let q ∈ R ∗ . Show by recurrence that if the terms of the sequence ( u n ) ver- ifies u n +1 u n = q for any natural number n , then for any n ∈ N , we have : u n = u 0 × q n https://chingmath.fr chapExoCorrec/4582 sacados/4582 sacados/57 chapExoCorrec/7593 sacados/7593 chapExoCorrec/7594 sacados/7594 chapExoCorrec/4577 sacados/4577 chapExoCorrec/4618 sacados/4618 chapExoCorrec/9494 sacados/9494 chapExoCorrec/4574 sacados/4574 chapExoCorrec/7596 sacados/7596 chapExoCorrec/4573 sacados/4573 sacados/48
E.9490 Consider the sequence v n defined ex-plicitly, for any n ∈ N , by the relation as a function of rank n : v n = 2 × 3 n Justify that the sequence v n is a geometric sequence. Give the reason for this sequence. E.7718 Consider the sequence v n defined for any positive integer or zero ( n ∈ N ) geometric whose value of the following two terms is known : v 3 =256 ; v 8 =781.25 Determine the characteristic elements of the terms in the se-quence v n . E.7749 Consider the sequence v n defined for any positive integer or zero ( n ∈ N ) geometric whose value of the following two terms is known : v 2 =250 ; v 7 =81.92 Determine the characteristic elements of the terms in the se-quence v n . E.9489 Consider the sequence v n geometric, defined for any n ∈ N , and for which the values of the two terms are known : v 2 = 2 ; v 5 = 54 1 Determine the characteristic elements of the suite v n . 2 Give the recurrence formula, then the explicit formula characterizing the sequence v n E.9491 Consider the sequence v n geometric, where n ∈ N , for which the values of the following two terms are known : v 3 = 4 ; v 8 = 243 8 1 Determine the characteristic elements of the suite v n . Justify your approach. 2 Give the explicit formula defining the value of a term in the sequence v n as a function of its rank. E.4608 Consider the sequence u n de-fined for any n ∈ N and whose term of rank n is defined by: u n = 7 × 4 n − 2 × 3 n 1 Show that the sequence u n verifies the relationship : u n +2 = 7 · u n +1 − 12 · u n 2 a Establish identity: u n +2 − 3 u n +1 = 4 · u n +1 − 3 u n b We define the sequence v n n ∈ N by the relation: v n = u n +1 − 3 · u n Determine the nature and characteristic elements of the sequence v n . 6. Geometric sequences: recognizing E.9485 Consider the sequence v n defined for any n ∈ N and whose first terms are: v 0 = 8 ; v 1 = 4 ; v 2 = 2 ; v 3 = 1 2 Justify that the sequence v n is not a geometric sequence. E.9487 Consider the sequence v n defined for any n ∈ N and whose first terms are: v 0 = 108 ; v 1 = 36 ; v 2 = 12 ; v 3 = 2 Justify that the sequence v n is not a geometric sequence. 7. Geometric sequences: modeling E.7202 A website offers its subscribers movies to download. When it opened, 500 films were offered and each month the number of films offered to subscribers increased by 6 % . We denote by u n the number of films offered where n denotes the number of months since the site opened. 1 Calculate u 1 and u 2 and give the result rounded to unity. In the remainder of the exercise, it is assumed that the se-quence u n is a geometric sequence of prime 500 : 2 Express u n as a function of n . https://chingmath.fr chapExoCorrec/9490 sacados/9490 chapExoCorrec/7718 sacados/7718 chapExoCorrec/7749 sacados/7749 chapExoCorrec/9489 sacados/9489 chapExoCorrec/9491 sacados/9491 chapExoCorrec/4608 sacados/4608 chapExoCorrec/9485 sacados/9485 chapExoCorrec/9487 sacados/9487 chapExoCorrec/7202 sacados/7202
E.2025 The evolution of a disease between 1987 and 2001 is modeled by a function f , whose graphical representation is given below : 1 Draw the table of variations of this function on the inter-val [1987 ; 2001] . 2 Over what period is there an increase in the number of new cases of disease? 3 What is the maximum number of new cases reported? In which year? 4 The number of new cases between 1998 and 2001 is shown in the following table : Année 1998 1999 2000 2001 Nombre de nouveaux cas 1908 1777 1668 1552 By what percentage does the number of new cases vary between 1998 and 1999, between 1999 and 2000, and be-tween 2000 and 2001? Round percentages to the nearest whole number. 5 It is assumed that, from 2001 onwards, the number of new cases of disease in the year, 2001+ n a Express u n +1 in terms of u n . What is the nature of the sequence ( u n ) ? b Express u n as a function of n . c How many new cases of disease can we estimate for 2003? For 2004? E.7567 A job seeker is offered two options : An initial salary of 1150 euros per month and a raise of 5 % per month. We note a n the sequence of these monthly incomes with this proposal. An initial salary of 1200 euros per month and an increase of 3 % per month. We note b n the continuation of these monthly incomes with this proposal. 1 Give the nature and characteristic elements of each of the sequences a n and b n . 2 Complete the table below by rounding the values of the terms to the nearest hundredth. n 0 1 2 3 4 a n b n 3 a At the end of 5 ième months, which proposal offers the most advantageous salary? b Based on the amount received at the end of five months, which proposal is the most advantageous? E.1890 Part A In the figure below, ABCDEF is a regular hexagon with cen-ter O and A , B , C ’, D , E , F are the middles of the sides of this hexagon. We admit that A B C D E F is also a hexagon with center O . On the other hand, recall that any triangle formed by the center and two consecutive vertices of a regular hexagon is an equilateral triangle. 1 We assume that : OA = 3 2 · OA . Show that : A B AB = 3 2 . 2 We note s , s , a and a the respective areas of the triangle OAB , the triangle OA B , the hexagon ABCDEF and the hexagon A B C D E F . a Prove that : s s = 3 4 . b Express a in terms of a . https://chingmath.fr sacados/2025 Année019881990199219941996199820002002Nombre de cas100020003000400050006000 chapExoCorrec/7567 sacados/7567 sacados/1890 France - Septembre 2006 - 9 points ABCDEFA0B0C0D0F0E0
Part B In this part, the results from Part A can be used even if they have not been demonstrated. In the figure in the appendix, A 0 B 0 C 0 D 0 E 0 F 0 is a regular hexagon whose side A 0 B 0 measures 8 cm and A 1 , B 1 , C 1 , D 1 , E 1 , F 1 are the middles of the sides of this hexagon. More generally, for any natural number n , note A n +1 , B n +1 , C n +1 , D n +1 , E n +1 , F n +1 the middles of the sides [ A n B n ] , [ B n C n ] ,. . . , [ F n A n ] of the hexagon A N B n C n D n E n F n . 1 On the figure on the appendix sheet to be returned with the copy, draw the hexagons A 3 B 3 C 3 D 3 E 3 F 3 and A 4 B 4 C 4 D 4 E 4 F 4 . 2 We note c n the side in centimetres of the hexagon A n B n C n D n E n F n . a Prove that ( c n ) is a geometric sequence of reason 3 2 . b Deduce the expression of c n as a function of n . c Determine the exact value of length A 4 B 4 . 3 Note a n the area in cm 2 of the hexagon A n B n C n D n E n F n . a What is the nature of the sequence ( a n ) ? Show that for any natural number n , we have : a n = a 0 · 3 4 n . b Check that the area of A 4 B 4 C 4 D 4 E 4 F 4 is less than a third of that of A 0 B 0 C 0 D 0 E 0 F 0 . 8. Arithmetic and geometric sequences E.4576 The sequences of this exercise are defined for a positive integer or zero rank n ( n ∈ N ) : 1 Determine the nature of each of the sequences below and its characteristic elements : a a 0 =3 ; a n +1 = a n − 2 b b 0 =5 ; b n +1 =2 · b n c c 0 = 1 2 ; c n +1 = 3 4 · c n d d 0 = − 1 ; d n +1 = d n +1 2 Determine the nature of each of the sequences below and its characteristic elements : a e n = 3 n − 2 b f n = 2 × 3 n c g n = 5 2 n d h n = 1 − n E.4710 Consider the two sequences u n and v n , defined for ranks n positive or zero ( n ∈ N ) whose first terms are given below : u n : (2 ; 6 ; 9 ; 12 ; : : : ) v n : 54 ; 6 ; 2 9 ; 2 81 ; : : : Justify that each of these sequences can be neither arithmetic nor geometric. E.37 Here are the first 7 terms of four sequences : 1 13 ; 14.5 ; 16 ; 17.5 ; 19 ; 20.5 ; 22 2 300 ; 278 ; 266 ; 254 ; 244 ; 232 ; 220 3 2 ; 6 ; 18 ; 44 ; 132 ; 396 ; 1188 4 0.0625 ; 0.125 ; 0.25 ; 0.5 ; 1 ; 2 ; 4 Determine among these sequences : those that can be arithmetic or geometric in nature. those that are neither arithmetic nor geometric. E.187 Consider the sequence ( u n ) n whose first terms are known : Rank n 0 1 2 3 Term value u n 2 4 8 10 1 Show that the sequence ( u n ) n is not an arithmetic se-quence. 2 Show that the sequence ( u n ) n is not a geometric se-quence. https://chingmath.fr ABCDEFA2B2C2D2E2F2A1B1C1D1F1E1 chapExoCorrec/4576 sacados/4576 chapExoCorrec/4710 sacados/4710 chapExoCorrec/37 sacados/37 sacados/187
E.4578 Consider the two sequences u n and v n définies by recurrence by relations where the ranks n of their terms are positive or zero integers ( n ∈ N ) : u n +1 = u n + 0.75 ; u 0 = 2 v n +1 = 2 · v n ; v 0 = 0.125 1 What are the natures of the sequences u n et v n ? The characteristic elements of these two sequences will be specified. 2 Complete the following table with the values of the se-quences rounded to the nearest tenth : n 0 1 2 3 4 5 6 7 8 9 u n v n 3 Place the points ( n ; u n ) and ( n ; v n ) représentant these two sequences in the frame O ; I ; J ci below : E.9492 1 Consider the sequence ( v n ) n arithmetic with first term u 0 =3 and reason 1.5 . Give the value of the term v n as a function of the rank n . 2 Consider the sequence ( w n ) n geometric with first term w 0 =2 and reason 2. Give the value of the term w n as a function of the rank n . 9. Arithmetic and geometric sequences: modeling E.175 As part of their T . P . E . (Travaux Person-nels Encadrés) , two first-year high school students want to study the evolution of the frog population in the local pond. According to the local environmental club, this population is on the verge of extinction, and the club members are worried. To carry out their study, the two high-school students initially have only the following two surveys carried out by the club : Statement date 1 er novembre 2002 1 er novembre 2003 Frog population 1000 950 Rank n of year 0 1 To practice forecasting, the two high school students model the evolution of the frog population using a sequence. Part A The two high school students hypothesize that an arithmetic sequence can be used to model the evolution of the frog pop-ulation. They note this sequence ( u n ) where u 0 is the frog population on 1 er November 2002 and more generally, u n is the frog population on 1 er November (2002+ n ) . 1 Calculate the reason r of the sequence ( u n ) . 2 According to this model, what would the frog population be on 1 er November 2005? The 1 er November 2012? Le 1 er novembre (2002+ n ) ? 3 Determine the year when the frog population will have totally disappeared according to this model. 4 The two high school students receive the survey taken on 1 er November 2004: 903 frogs. Does this new result confirm their hypothesis? Part B Continuing their thinking, the two high school students ask themselves whether a geometric sequence ( v n ) would model the evolution of the frog population. v 0 would then be the frog population on 1 er November 2002 and more generally, v n the frog population on 1 er November (2002+ n ) . 1 a Verify that the sequence ( v n ) has reason 0.95 . b Explain why this new model seems a better fit. 2 a What would then be, to the nearest integer, the frog population on 1 er November 2005? b For any natural number n , write v n as a function of n . c Deduce, to the nearest integer, what the frog popula-tion would be on 1 er November 2012? 3 The two high school students also wonder from what date the frog population in the pond would be reduced to less than two frogs. Answer this question with the help of the calculator, giving the results that lead to the conclusion. https://chingmath.fr chapExoCorrec/4578 sacados/4578 23456789I23456789JO sacados/9492 sacados/175
E.7499 Scientists are studying a bacterial cul-ture containing two strains, which we will call A and B . At the start of the experiment (at time ˇ0ı) , there are 200 bacteria of strain A and 300 bacteria of strain B . Scientists observe the following changes : every minute, the population of bacteria A increases by 10 % , while that of strain B decreases by 20 bacteria. 1 Complete the table below : (Round the data to the nearest whole number.) . 2 Let n be a natural number ( n ∈ N ) , we note a n the popu-lation of bacteria of strain A at time ˇ n min ı ; Complete the values of the terms a n : a 0 =200 ; a 1 = ::: ; a 2 = ::: ; a 3 = ::: 3 Let n be a natural number ( n ∈ N ) , we note b n the popu-lation of bacteria of strain B at time ˇ n min ı ; Complete the values of the terms b n : b 0 =300 ; b 1 = ::: ; b 2 = ::: ; b 3 = ::: E.40 A company director hires a young technician and offers him two types of compensation starting January 1, 2000. 1 First type of compensation. For this first year, 2000, he will receive 22 400 euros, fol-lowed by a steady annual increase of 750 euros. Note u 0 the salary in euros for the year 2000, u 1 the salary in euros for the year 2001, and, in general, u n the salary in euros for the year 2000+ n ( for n a natural number) . a Calculate the annual salaries u 1 for the year 2001 and u 2 for the year 2002. b Specify the nature of the sequence ( u n ) by stating its reason c Show that : u n =22 400+750 · n 2 Second type of sequence For the year 2000 , he will also receive 22 400 euros, but thereafter, each year, an increase of 3% compared to the previous year. In this case, let v n denote the amount in euros of the compensation for year 2000+ n (for n , a natural number) . a Calculate the annual salaries v 1 for year 2001 and v 2 for year 2002 . b Show that v n +1 =1 ; 03 · v n for all n . Determine the na-ture of the sequence ( v n ) c Determine the expression of v n in terms of n . 3 Comparison a Calculate the annual salary for 2008 in each of the two cases. b For 2008, identify the most advantageous type of com-pensation. https://chingmath.fr chapExoCorrec/7499 sacados/7499 1234567ABCTemps(enmin)PopulationdelasoucheAPopulationdelasoucheB020030012345 chapExoCorrec/40 sacados/40
E.49 A company offers a prospective employee two types of contracts : Option 1 : the starting salary will be 1200 e and an in-crease of 50 e will be applied applied to their salary each year. Formula 2 : the starting salary will be 1100 e and an increase of 5 % will be applied to their salary each year. We note u n your salary of the first type n year after you start, u n your salary with formula 1 and v n with formula 2. 1 Determine the value of u n and v n based on the index n . 2 What can be said about the growth of these two se-quences? 3 a Complete the table below using the first 10 terms of each sequence. b Determine when it is preferable to choose the second formula. 4 Using spreadsheet software, plot the curves of these two sequences and find your result. 5 Using the same software, provide the formula that will enable the future employee to earn the most money after 10 years? 15 years? 20 years? E.4543 The Mandine company hires Arthur on January 1 er , 2009, with a salary of 1525 e and offers him two types of advancement : Every January 1 er , his salary will increase by 32 e . Every January 1 er , his salary increases by 2 % . 1 Complete the following table, rounding the values to the nearest tenth : Year 2009 2010 2011 2012 Advancement A Advancement B Year 2013 2014 2015 2016 Promotion A Promotion B 2 From which year will Arthur have a higher salary by choosing promotion B ? E.4555 In an imaginary country rated I , there is a capital city P and a group of villages V . On January 1 er , 2002, P and V had 200 000 and 300 000 inhab-itants, respectively. Each year, the population of P increases by 10 % , while that of V decreases by 20 000 inhabitants. 1 a On January 1 er , 2002, what percentage of the popu-lation of P did the population of I represent?? b Calculate the population of P , then that of V , and fi-nally that of I on January 1 er , 2003. What percentage does the population of P represent in relation to that of I ?? c Complete the table below, rounding to the nearest whole number: 2 n denotes a positive integer or zero. ( n ∈ N ) . We denote p n as the population of P on January 1 er (2002+ n ) ; as follows : p 0 =200 000 . We denote v n as the population of V on 1 er January (2002+ n ) ; as follows : v 0 =300 000 . a Express p n +1 in terms of p n . b Express v n +1 in terms of v n . 3 Complete the two diagrams below : a b https://chingmath.fr sacados/49 12345ABCnunvn1234 chapExoCorrec/4543 sacados/4543 chapExoCorrec/4555 sacados/4555 1234567ABCDAnnéePopulationdePau1janvierPopulationdeVau1janvierPopulationdeIau1janvier2002200000300000 p0×:::p1×:::p2×:::p3×:::p4×:::×:::×:::×::: v0−:::v1−:::v2−:::v3−:::v4−:::−:::−:::−:::
E.173 A company offers a future employee two types of contract : Formula 1 : his first salary will be 1200 e per month and an increase of 50 e will be applied to his salary each year. Formula 2 : his first salary will be 1100 e per month and an increase of 5 % will be applied to his salary each year. We note u n your salary of the first type n year after your start, u n your salary with formula 1 and v n with formula 2. 1 Complete the table below : 2 a From the previous table, place, in the reference frame below, the points with coordinates : ( n ; u n ) ; ( n ; v n ) b Connect these points. 10. Sum of terms E.32 Part A A company’s production can be modeled by an arithmetic se-quence ( u n ) such that, for any nonzero natural number n , u n denotes the number of devices produced in year n . In year 1 re , production is 7 500 devices ; we have : u 1 = 7500 In year 6 e , production is 12 000 units ; we have : u 6 = 12 000 1 Show that the common ratio of the sequence ( u n ) is 900 . 2 For any nonzero natural number n , express u n in terms of n . 3 After how many years will production have exceeded three times the initial production? Part B Another company produces 1 re units in year 7 500 . Let n , v n denote, for any nonzero natural number, the number of units produced in year n . We therefore have v 1 =7500 . This company’s annual production increases by 10 % each year. 1 Calculate v 2 and v 3 . 2 Show that the sequence ( v n ) is a geometric sequence. Give its common ratio. 3 For any nonzero natural number n , express v n in terms of n . 4 After how many years will production have exceeded three times the initial production? 5 How many devices will the company have produced in 13 years? Reminder : for q =1 : 1 + q + q 2 + · · · + q n = 1 − q n +1 1 − q 11. Study of sequences with an intermediate sequence E.50 We wish to study the sequence ( u n ) with first term u 0 =5 defined by the following recurrence relation: u n +1 = 1 3 · u n + 4 Let: v n = u n − 6 . 1 Show that the sequence ( v n ) is a geometric sequence, specifying its first term and common ratio. 2 Express v n in terms of the rank n . 3 Deduce the expression for u n in terms of n . E.61 Consider the sequence ( u n ) defined by the following recurrence relation: u 0 = 8 ; u n +1 = 1 2 · u n − 5 We wish to determine the expression of ( u n ) as a function of its rank: 1 a We define the sequence ( v n ) by: v n = u n +10 . Show that the sequence ( v n ) is a geometric sequence whose first term and reason are to be specified. b Establish the expression of the term v n as a function https://chingmath.fr sacados/173 123456789ABCnunvn12345678 01234567891011001300150017001900 chapExoCorrec/32 sacados/32 Liban - 2005 - Obligatoire - 7 points chapExoCorrec/50 sacados/50 chapExoCorrec/61 sacados/61
of rank n . 2 Deduce an expression for the term u n as a function of n . E.58 On 1 er January 2004, I have a sum u 0 of 1 000 e in my account paying compound interest at 2 % per year. We note : u 0 =1000 . Interest is paid annually on December 31. I decide that starting in 2005 to withdraw each year 100 e on 1 er January. I call u n the balance at 1 er January of the year (2004+ n ) after my withdrawal from 100 e . 1 a Calculate the balances u 1 and u 2 of this account. b Is the sequence of general term u n arithmetic? Is it geometric? c Show that for any n natural number: u n +1 = 1.02 · u n − 100 2 a We pose, for any n natural number, v n = u n − 5 000 . Calculate v 0 . b Show that for any n : v n +1 =1.02 · v n c Express the general term v n of the sequence ( v n ) in terms of n . 3 a Deduce the expression of u n as a function of n . b Calculate u 10 rounding to the nearest 0.01 . c As of 1 er January of which year will my account have a negative balance for the first time? E.63 Let ( u n ) be defined by its first term u 0 = 6 and by the recurrence relation: u n +1 = 1 4 · u n + 3 ( n is a 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 . E.47 Thistles are invading a lawn in two different ways. This Sunday, June 13, they cover 300 m 2 of the lawn. Each week, the area of the lawn invaded by thistles increases by 4 % due to root proliferation and by 13 m 2 due to seeds blown in from neighboring gardens. Let u n be the area of the lawn, and m 2 the area overrun by thistles after n weeks. We therefore have : u 0 =300 . 1 Calculate u 1 , u 2 , and u 3 . 2 Prove that for any natural number n : u n +1 = 1 ; 04 × u n + 13 . 3 The sequence ( v n ) n 0 is defined by: v n = u n +325 . a Prove that : v n +1 =1 ; 04 · v n . b Conclude that the sequence ( v n ) is a geometric se-quence, specifying its first term and common ratio. 4 Express v n in terms of n , and conclude that : u n = 625 × (1 ; 04) n − 325 . 5 After how many weeks will the thistles have taken over more than 700 m 2 of the lawn? E.55 Pierre invests 200 euros in an ac-count with his bank on January 1, 2003. The bank pays an annual interest rate of 4 % . We note u 0 the initial amount of the account, so u 0 =200 and u n the amount at 1 er January of the year (2003+ n ) , n being a natural number. 1 Calculate u 1 , u 2 and u 3 . Round to the nearest euro cent. 2 Express u n +1 as a function of u n 3 We define a new sequence ( v n ) by posing, for any natural number n : v n = u n +5000 . a Calculate the first three terms of the sequence ( v n ) . b Prove that the sequence ( v n ) is geometric and specify its reason. c Then express v n as a function of n , then deduce that : u n = 5200 × (1.04) n − 5000 4 How many years will Pierre have to wait, to have at least 3 000 of euros in this account? For this question, we can use the neperian logarithm function noted ln https://chingmath.fr chapExoCorrec/58 sacados/58 chapExoCorrec/63 sacados/63 chapExoCorrec/47 sacados/47 Liban - 2004 - 6 points - obligatoire chapExoCorrec/55 sacados/55
E.64 Two friends, Agnès and Bénédicte win 2 000 e in a game. They divide this sum into two equal parts. Part A Agnès, who already has 3 000 e of savings, adds her 1 000 e to her savings and places the total in a savings passbook that earns 3.5% interest per year (compound interest) . We note u 0 the capital invested ( u 0 =4000 ) , u 1 the capital acquired after one year, and more generally u n le capital ac-quired after n années. 1 Calculate u 1 and u 2 . 2 Express u n +1 en function of u n . Deduce the nature of the sequence ( u n ) . 3 Express the general term u n as a function of n . 4 What will be the capital obtained at the end of 6 years? (The result will be rounded to the nearest cent) Part B Bénédicte chooses a savings account with a monthly rate of 0 ; 25 % and chooses to add to it at the end of each month the sum of 50 e . The interest earned is capitalized at the end of each month. We note v 0 the capital invested ( v 0 =1000 ) , v 1 le capital ac-quired at the end of one month, and more generally v n le capital acquired at the end of n mois. 1 Calculate v 1 and v 2 (the result will be rounded to the nearest cent) . Check that : v 3 =1157.89 . 2 For any natural number n , express v n +1 en function of v n . 3 Consider the sequence ( w n ) définie for any natural num-ber n par w n = v n +20000 . a Prove that the sequence ( w n ) is a geometric sequence with common ratio 1.0025 . Specify w 0 and express the general term w n in terms of n . Deduce v n as a function of n . b Calculate the capital acquired by Bénédicte after 6 years (i.e., 72 months) . (Round the result to the near-est cent) E.51 Consider the sequence defined for any natural number n by: u 0 = − 3 2 ; u n +1 = 2 3 · u n − 1 for all n ∈ N 1 a Calculate u 1 and u 2 . b Is the sequence ( u n ) n ∈ N arithmetic? geometric? 2 We pose v n = u n +3 . Show that the sequence ( v n ) n ∈ N is a geometric sequence ; determine its reason and first term. 3 Give the expression of v n , then of u n as a function of n . 4 Deduce the limit of the sequence ( u n ) n ∈ N . 5 We pose : s n = v 0 + v 1 + · · · + v n − 1 Express s n as a function of n and derive lim n ↦→ + ∞ s n . E.38 Pierre makes an investment in his bank by paying into an account 200 euros, every first of Jan-uary from 01 = 01 = 2003 . The bank remunerates this account at an annual rate of 4 % . We note u 0 the initial amount of the account, so u 0 and u n the amount on 1 er January of the year (2003+ n ) , n being a natural number. 1 Calculate u 1 , u 2 and u 3 . Round to the nearest euro cent. 2 Express u n +1 as a function of u n . 3 We define a new sequence ( v n ) by posing, for any natural number n : v n = u n +5000 . a Calculate the first three terms of the sequence ( v n ) . b Prove that the sequence ( v n ) is geometric and specify its reason. c Then express v n as a function of n , then deduce that : u n =5200 × (1.04) n − 5000 . 4 How many years will Pierre have to wait, to have at least 3000 euros in this account? For this question, we can use the neperian logarithm func-tion noted ln . E.41 Part A : Study of a series Let ( u n ) be the sequence defined by: u 0 = 1 500 000 u n +1 = 1.013 · u n + 1 300 for any natural number n 1 Calculate u 1 and u 2 2 We pose for any natural number n : v n = u n +100 000 a Calculate v 0 . b Show that, for any natural number n ; v n +1 = 1.013 · v n . Deduce the nature of the sequence ( v n ) . c Determine v n as a function of n . Deduce that : u n =1 600 000 · 1.013 n − 100 000 . d Calculate u 18 . The result will be rounded to the near-est integer. Part B: Application For this part, all numerical results will be rounded to the nearest integer A study of the population of a département reveals the fol-lowing information : population is estimated at 1 500 000 inhabitants in 2002 the natural growth rate is 1.3% per year, the migratory flow (difference between the number of peo-ple entering the department and the number leaving) is estimated at 1.300 inhabitants per year. It is estimated that these figures will remain constant over the years 1 Determine the estimated population of this department in 2003 and 2004 2 We pose w 0 =1 500 000 . For any natural number n , we denote by w n an estimate of the number of inhabitants of this department during the year (2002+ n ) a Check that w n +1 =1.013 · w n +1300 for any natural number n . b Using the part A , deduce an estimate of the population https://chingmath.fr chapExoCorrec/64 sacados/64 Antilles - 2004 - 7 points - obligatoire chapExoCorrec/51 sacados/51 France - Juin 2001 - 5 points chapExoCorrec/38 sacados/38 chapExoCorrec/41 sacados/41 Canada -- 2002 -- 5 points -- au choix
of this department in 2020. E.52 Part A Let ( u n ) be the sequence defined by its first term u 0 =0.7 and by the recurrence relation: u n +1 = 2 · u n − 0.4 ( n natural number) 1 Calculate u 1 , u 2 , u 3 . 2 Let ( v n ) be the sequence defined for any natural integer n by: v n = u n − 0.4 . a Calculate v 0 . b Show that : v n +1 =2 · v n c Deduce the nature of the sequence. d Express v n then u n as a function of n . Part B The Titius-Bose rule (known around 1770) is used to approx-imate the distance to the Sun of most planets in the solar system. To do this, we take as our unit the distance from the Earth to the Sun, which is worth around 150 million kilo-meters. This distance is called the astronomical unit (u.a.) . Thus, 1 u . a . ≈ 15 × 10 7 km . In modern writing, the Titius- Bode law is expressed by the following formula : u n = 0.4 + 0.3 × 2 n where for a given planet u n is that planet’s distance from the Sun (in a.u.) and n is the planet’s rank, defined in the table below : Planet Venus Earth Mars (Ceres)* Jupiter Saturne Uranus Rank n 0 1 2 3 4 5 6 (*) The gap observed between the orbits of Mars and Jupiter was filled in 1801 by the discovery of the planet Ceres, then later of thousands of asteroids 1 Copy and complete the following table : n 0 1 2 3 4 5 6 u n 2 a Calculate the approximate distance to the Sun of the planet Uranus (we’ll give the result in millions of kilometers) . b Calculate the rank of the planet whose approximate distance from the Sun is 780 million kilometers. Which planet is it? 12. Share E.9493 Paul invests the sum of 500 e in the bank. The bank gives him 5 % interest per year: 1 Donner les caractéristiques de la suite symbolisant l’évolution de la somme placée à la banque. 2 Give this value as a function of the number of years n after depositing the money. 3 After how long will he have more 700 e . 13. Unclassified financial years E.179 1 2 3 4 6 5 7 8 9 10 11 12 13 14 15 16 We imagine that we arrange, according to the model above, continuing it, the sequence of integers up to 2 500 . Rows are numbered from the top, boxes from the left. Each number is thus identified by its line number, then by its box number on that line. Example : 15 is the 4 e line and 6 e box Part A In this part, we’ll look at some properties due to the layout used. 1 Number of boxes per line . We note u n the number of squares that make up the line n . a Donner u 1 , u 2 , u 3 , u 4 , u 5 and u 6 . b Express u n +1 in terms of u n . Deduce the nature of the sequence ( u n ) . c Show that : u n =2 n − 1 . 2 Last number in each line . We note d n the last num-ber of the line n . Give d 1 , d 2 , d 3 , d 4 , d 5 and d 6 . It will be assumed throughout the rest of the problem that for n 1 , d n = n 2 . 3 First number in each line . We note o n the first num-ber in line n . a Donner p 1 , p 2 , p 3 , p 4 , p 5 and p 6 . b Express p n as a function of d n − 1 . c Deduct : p n = n 2 − 2 n + 2 . Part B In this part, we look for the place of 1 500 in the table (its https://chingmath.fr chapExoCorrec/52 sacados/52 sacados/9493 sacados/179 Asie -- Juin 2005 - 11 points
row and cell numbers) 1 We first look for the line number in which the number 1 500 appears. Determine an integer n such that n 2 < 1 500 < ( n +1) 2 , and conclude. 2 Below, the row containing 1 500 has been extracted from the table. Specify the four missing values, i.e. the line number, the numbers of the first and last boxes and the number of the box containing 1 500 . E.7810 L’indice de référence des loyers (IRL) is used as a basis for revising rents for empty or furnished accommodation. It sets ceilings on the annual rent increases that landlords can demand. Source : http://service-public.fr A dwelling is rented in 2018 for a rent of 814 e and whose increase is set at 0.5 % per year. We note u n the amount of rent to the year 2018+ n . 1 What is the nature of the sequence u n ? Give the char-acteristic elements of the sequence u b . 2 a Give the expression of the terms of the sequence u n as a function of n . b Determine the amount of rent in 2030 . 3 Using the calculator, determine the year in which rent first exceeds 900 e . E.6126 Consider the two sequences u n and v n defined recursively by the relations : u 0 =2 ; u n +1 = u n +0.75 for all n ∈ N v 0 =0.125 ; v n +1 =2 · v n for all n ∈ N 1 What are the natures of the sequences u n and v n ? 2 Complete the following table with the values of the se-quence rounded to the nearest tenth : n 0 1 2 3 4 5 6 7 8 9 u n v n 3 Place the points ( n ; u n ) and ( n ; v n ) representing these two sequences on the graph O ; I ; J below : E.6127 1 Consider the sequence u n defined explicitly by the re-lation as a function of rank n : u n = 3 · n + 2 for all n ∈ N Justify that the sequence u n is an arithmetic sequence. Give the reason for this sequence. 2 Consider the sequence v n defined explicitly by the re-lation as a function of rank n : v n = 2 × 3 n for all n ∈ N Justify that the sequence v n is a geometric sequence. Give the reason for this sequence. https://chingmath.fr chapExoCorrec/7810 sacados/7810 chapExoCorrec/6126 sacados/6126 23456789I23456789JO chapExoCorrec/6127 sacados/6127
E.8356 Consider the two sequences u n and v n where : ( u n ) an arithmetic sequence with first term 3 and com-mon ratio 0.5 ; ( v n ) a geometric sequence with first term 1 and common ratio 1.3 . 1 Complete the table below with the terms of these two sequences, rounding the values to two decimal places : n u n v n 0 1 2 3 4 5 6 7 8 9 2 Place the points ( n ; u n ) and ( n ; v n ) on the graph be-low : E.2387 Consider the sequence u n n ∈ N de-fined by the explicit formula : u n = 5 + 2 × n for any natural number n . 1 Express the value u n − 3 as a function of n . 2 Give the simplified form of u n − 3 + u 3 . 3 Give the simplified form of u n − 5 + u 5 . 4 Let k and n be two integers such that k n . Show that u k + u n − k has its value independent of k . E.5106 Consider the function f defined by: f ( x )= 9 6 − x whose representative curve C f is given in the reference frame O ; I ; J orthonormal below : We define the sequence u n by: u 0 = − 3 ; u n +1 = 9 6 − u n for all n ∈ N 1 Algebraically determine the value of the first four terms of the sequence u n . 2 Graphically, place on the x-axis the first six terms of the sequence u n . https://chingmath.fr chapExoCorrec/8356 sacados/8356 23456789I234567891011JO chapExoCorrec/2387 sacados/2387 chapExoCorrec/5106 sacados/5106 -3-2-1234I234JOCf