Grade 12 - Comp. / Numerical sequences 47 exercises (100% corrected)

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1. A little further: arithmetic-geometric sequences E.2408 Consider the sequence u n n ∈ N defined by the relation: u 0 = 8 ; u n +1 = 1 2 u n − 5 for any natural number n . 1 Let v n n ∈ N be the sequence defined by: v n = u n + 10 for all n ∈ N a Show that the sequence v n verifies the following re-lationship for any natural number n : v n +1 = 1 2 · v n b Give the nature of the sequence ( v n ) and its character-istic elements. c Give the explicit formula giving the expression of the term v n as a function of its rank n . 2 From the previous questions, deduce the explicit formula for the sequence u n . E.6426 Let u n be defined by its first term u 0 and, for any natural number n , by the relation: u n +1 = a · u n + b ( a and b non-zero real numbers such that a =1 ) For any natural number n : v n = u n − b 1 − a Demonstrate that, the sequence v n is geometric of reason a . E.2409 Consider the sequence ( u n ) defined by: u 0 = 5 ; u n +1 = 1 3 u n + 4 for all n ∈ N 1 Let v n be the sequence defined by the relation: v n = u n − 6 for n ∈ N a Establish that the sequence v n is a geometric se-quence whose first term and reason are to be specified. b Give the expression of the term v n as a function of its rank n . 2 Deduce the expression of u n as a function of n . 2. Reminders E.7544 1 Aquarium technicians want to set up the automatic dis-penser for a product designed to improve water quality in a pond. At the start of the test, the concentration of the product in this pond is 160 mg · ‘ − 1 . It is estimated that the concentration of the product de-creases by approximately 10 % per week. Give the concentration of this product after 8 weeks. 2 Technicians set up a vending machine so that it dis-charges a certain amount of product each week. After study, he observes that with the use of this machine, the concentration of this product increases by 5 % per week. Starting the measurements with a concentration of 120 mg · ‘ − 1 , what will be the measurement of the con-centration of this product after 8 weeks? Concentrations obtained will be rounded to the nearest tenth. E.7543 1 Consider the sequence u n , defined on N , geometric with first term 3 and reason 2.5 . a Determine, rounded to the nearest thousandth, the value of terms u 5 and u 7 . b Give the limit of the terms of the sequence u n when n tends to + ∞ . 2 Consider the suite v n , defined on N , geometric with first term 7 and reason 0.5 . a Determine, rounded to the nearest 10 − 5 , the value of the terms v 5 and v 7 . b Give the limit of the terms of the sequence v n when n tends to + ∞ . E.7542 Each of the expressions below is used to define the terms of a suite u n geometric, defined on N : a u 0 = 3 ; u n +1 = 2 · u n b u n = 2 n c u 0 = − 2 ; u n +1 = u n 2 d u n = 3 × 0.2 n In each question, give the characteristic elements of the se-quence u n . E.7541 Among the four propositions made below, only one is cor-rect. Which one? Justify. Consider the sequence u n , defined for any natural number n , geometric with first term 4 and reason 3 : The sum S of the 10 first terms of the sequence defined by: S = u 0 + u 1 + · · · + u 9 has the value : a S = 2 · 3 9 − 1 b S = 2 · 3 10 − 1 c S = 4 · 3 9 − 1 d S = 4 · 3 10 − 1 3. Modeling a problem https://chingmath.fr chapExoCorrec/2408 sacados/2408 chapExoCorrec/6426 sacados/6426 chapExoCorrec/2409 sacados/2409 chapExoCorrec/7544 sacados/7544 chapExoCorrec/7543 sacados/7543 chapExoCorrec/7542 sacados/7542 chapExoCorrec/7541 sacados/7541
E.7040 On 1 er September 2015 , a school complex has 3 000 pupils. An internal statistical study has shown that every 1 er Septem-ber: 10 % headcount leaves ; 250 new students enroll. We seek to model this situation by a sequence u n where, for any natural number n , u n represents the number of students on 1 er September of the year 2015+ n . Justify that we can model the situation with the sequence u n such that u 0 =3 000 and, for any natural number n : u n +1 = 0.9 · u n + 250 E.7028 A company is interested in the number of 3 D screens it has sold since 2010 : Année 2010 2011 2012 Number of screens 3 D vendus 0 5 000 11 000 The number of screens 3 D sold by the company in the year (2010+ n ) is modeled by a sequence u n , arithmetic-geometric, of first term u 0 =0 . Recall that an arithmetic-geometric sequence verifies, for any natural number n , a recurrence relation of the form u n +1 = a × u n + b where a and b are two real numbers. 1 a Assuming u 1 =5 000 , determine the value of b . b Assuming further that u 2 =11 000 , show that for any natural number n , we have : u n +1 = 1.2 × u n + 5 000 2 a Calculate u 3 and u 4 . b In 2013 and 2014 , the company sold 18 000 and 27 000 screens respectively 3 D . Does the modeling seem relevant? E.7012 The company PiscinePlus , based in the south of France, offers annual maintenance contracts to private pool owners. The head of this company notes that, every year, 12 % ad-ditional contracts are taken out and 6 contracts terminated. He uses this observation to estimate the number of annual contracts to come. In 2015 , the company PiscinePlus counted 75 contracts sub-scribed. We model the situation by a sequence u n where u n repre-sents the number of contracts taken out with the company PiscinePlus in the year 2015+ n . Thus, we have u 0 =75 . 1 Estimate the number of maintenance contracts in 2016 . 2 Show that, for any natural number n , we have : u n +1 = 1.12 · u n − 6 4. Explicit formula E.7527 Let the sequence u n be defined by u 0 =150 and for any natural number n : u n +1 = 0.8 · u n + 45 Consider the sequence v n defined for any natural number n by: v n = u n − 225 1 Demonstrate that v n is a geometric sequence and spec-ify its first term and reason. 2 Deduce that for any natural number n : u n = 225 − 75 × 0.8 n E.7529 Consider the sequence u n de-fined for u 0 =27 500 and for any natural number n , we have : u n +1 = 1.04 · u n − 156 We seek to explicitly calculate the general term u n as a func-tion of n . To do this, note v n the sequence defined, for any natural number n , by: v n = u n − 3 900 1 Show that v n is a geometric sequence whose reason and first term should be specified. 2 Deduce that, for any natural number n : u n = 23 600 × 1.04 n + 3 900 E.7530 A plan to reduce greenhouse gas emissions (GHG) has been implemented on an industrial es-tate. It is estimated that, for companies already established on the site, the measures of this plan lead to a year-on-year reduction in emissions of 2 % and that, each year, new com-panies setting up on the site generate 200 tonnes of GHG equivalent CO 2 . In 2005 , this industrial estate emitted 41 thousands of tons of CO 2 in total. For any natural integer n , let u n be the number of thousand tonnes of CO 2 emitted in this industrial zone during the year 2005+ n . 1 Determine u 0 and u 1 . 2 Show that, for any natural number n , we have : u n +1 = 0.98 × u n + 0.2 3 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 geometric of reason 0.98 . Specify its first term. b Express v n as a function of n , for any natural number n . c Deduce that, for any natural number n : u n = 31 × 0.98 n + 10 5. Limit https://chingmath.fr chapExoCorrec/7040 sacados/7040 Extrait d'Asie Juin 2016 chapExoCorrec/7028 sacados/7028 chapExoCorrec/7012 sacados/7012 Extrait de Liban Mai 2016 chapExoCorrec/7527 sacados/7527 chapExoCorrec/7529 sacados/7529 chapExoCorrec/7530 sacados/7530 Extrait Liban Juin 2017
E.7033 A car rental company has a total of 10 000 cars for Europe at 1 er March 2015 . In order to maintain his fleet, he decides to resell, at 1 er March each year, 25 % of his car fleet and to buy 3 000 new cars. The number of cars in the agency is modeled using a sequence : For any natural number n , let u n be the number of cars in the fleet as at 1 er March of year 2015+ n . So we have : u 0 =10 000 1 Expliquer why for any natural number n : u n +1 = 0.75 · u n + 3000 2 Pour any natural number n , consider the sequence v n defined by: v n = u n − 12 000 a Montrer that the sequence v n is a geometric sequence of reason 0.75 . Specify its first term. b Express v n as a function of n . Determine the limit of the sequence v n . c Justify that, for any natural number n : u n = 12 000 − 2 000 × 0.75 n d En based on the answers given to the two previous questions, what can you conjecture about the number of cars this rental company’s fleet will have after a large number of years? E.7025 A website offers its subscribers movies to download. Due to a welcome offer, the number of subscribers at launch is 15 000 . Based on the first few months, we estimate that the number of customers subscribing to the site evolves according to the following rule : each month, 10 % customers unsubscribe and 2 500 new sub-scribers are registered. We note v n the estimated number of subscribers n months after opening, so we have v 0 =15 000 . 1 Justify that, for any natural number n , we have : v n +1 = 0.9 × v n + 2 500 2 Consider the sequence w n defined for any natural num-ber n by: w n = v n − 25 000 a Show that the sequence w n is geometric of reason 0.9 and specify its first term. b Deduce that, for any integer n : v n =25 000 − 10 000 × 0.9 n c Can this model be used to predict that the number of subscribers will stabilize over the long term? Justify the answer. E.7604 In 2015 , forests covered approximately 4 000 million hectares of land. It is estimated that this area decreases by 0.4 % each year. This loss is partly offset by natural or voluntary reforestation, which is estimated at 7.2 million hectares per year. Consider the sequence u n defined by u 0 =4 000 and, for any natural number n : u n +1 = 0.996 × u n + 7.2 1 Justify that, for any natural integer n , u n provides an estimate of the global forest area, in millions of hectares, for the year 2015+ n . 2 Copy and complete the algorithm below so that, at the end of its execution, the variable N has the value of the first year for which the total forest area covers less than 3 500 million hectares on earth. N ← 2015 U ← 4 000 ... ... ... 3 Consider the sequence v n defined for any natural num-ber n by: v n = u n − 1 800 a Prove that the sequence v n is geometric, then specify its first term and its ratio. b Deduce that for any natural number n , we have : u n = 2200 × 0.996 n + 1800 c According to this model, if the phenomenon continues, will the Earth’s forest cover eventually disappear? Jus-tify your answer. 6. Weir study E.7211 Every day, an association prepares and delivers meals to the homes of dependent people. In 2015 , 600 people subscribed to this service. To study its development, this association conducted a survey according to which the evaluation can be modeled as follows : Every year, 5 % subscriptions are not renewed. Each year, there are 80 new subscriptions to this service. This evolution can be studied using a sequence u n where u n is the number of subscribers during the year 2015+ n . We thus have, u 0 =600 and for any natural number n : u n +1 =0.95 · u n +80 Admit that the terms of the sequence u n admit as a func-tion of rank n the expression : u n = 1 600 − 1 000 × 0.95 n The size of the premises does not allow us to serve more than 1 000 meals. If this development continues at the same rate, will the asso-ciation one day have to consider expansion work? https://chingmath.fr chapExoCorrec/7033 sacados/7033 chapExoCorrec/7025 sacados/7025 chapExoCorrec/7604 sacados/7604 chapExoCorrec/7211 sacados/7211 Extrait Antilles-Guyanes Septembre 2016
E.7526 Let u n be defined by u 0 =150 and for any natural number n : u n +1 = 0.8 · u n + 45 1 Calculate u 1 and u 2 . 2 Here are two proposed algorithms : U ← 150 N ← 0 While U 220 U ← 0.8 × U+45 N ← N+1 End While Algorithm 1 U ← 150 N ← 0 While U<220 U ← 0.8 × U+45 N ← N+1 End While Algorithm 2 At the end of its execution, we are interested in the value of the variable N of the algorithm. a Only one of these algorithms allows the variable N , to be assigned, at the end of execution, the smallest nat-ural number n such that u n 220 . Specify which one, explaining why the other algorithm does not allow this. b What is the value of the variable N at the end of the execution of this algorithm? E.7528 A large university, experienc-ing rapid growth in enrollment, welcomed 27 500 students in September 2016 . The university president is concerned because he knows that, despite optimal management of the premises and distribution of students across the university’s various sites, he will not be able to accommodate more than 33 000 students. A statistical study allows him to develop a forecast model according to which, each year: 150 students drop out during the academic year (between September 1 er and June 30 ) ; the number of students enrolled at the start of the aca- demic year in September increases by 4 % compared to the previous month of June. For any natural number n , we note u n the number of stu-dents estimated according to this model at the start of the academic year in September 2016+ n , so we have u 0 =27 500 . 1 a Estimate the number of students in June 2017 . b Estimate the number of students at the start of the academic year in September 2017 . 2 Justify that, for any natural number n , we have : u n +1 = 1.04 · u n − 156 3 Copy and complete lines 3 , 4 , 5 , and 7 of the following algorithm so that, at the end of its execution, the vari-able n has a value equal to the year in which the number of students to be admitted will exceed the maximum ca-pacity of the institution ‘ .1 n ← 0 ‘ .2 U ← 27 500 ‘ .3 As long as U ... ‘ .4 n ← ... ‘ .5 U ← ... ‘ .6 End While ‘ .7 U ← U+ ... 4 a We run this algorithm step by step. Copy the following table and complete it by adding the necessary number of columns ; round the values of U to the nearest whole number. Initialization Step 1 . . . Valeur of n 0 . . . . . . Valeur of U 27 500 . . . . . . b Give the value of the variable n at the end of the exe-cution of this algorithm. 7. Solving inequalities (logarithm) E.7013 Consider the sequence u n de-fined by: u 0 = 75 ; u n +1 = 1.12 · u n − 6 for any n ∈ N We pose for any natural number n : v n = u n − 50 1 Show that the sequence v n is a geometric sequence. Specify its reason and first term. 2 Deduce the expression of v n as a function of n , then show that, for any natural number n , we have : u n 05.12 n +50 3 Solve in the set of natural numbers the inequation : u n > 100 . E.7016 Consider the sequence u n de-fined by: u 0 =4 ; u n +1 = 0.92 · u n + 8 for all n ∈ N For any natural number, we set : v n = u n − 100 1 Show that the sequence v n is geometric with common ratio 0.92 and calculate its first term v 0 . 2 Give the expression of v n in terms of n . 3 Deduce that, for any natural number n , we have : u n = 100 − 96 × 0.92 n . 4 Solve the inequality: u n 70 8. Reminders Sum of the terms of a sequence https://chingmath.fr chapExoCorrec/7526 sacados/7526 chapExoCorrec/7528 sacados/7528 chapExoCorrec/7013 sacados/7013 Extrait de Liban Mai 2016 chapExoCorrec/7016 sacados/7016
E.7261 For each question, determine the exact value of the sum, then, if necessary, its value rounded to two decimal places : a 1+4+4 2 + ··· +4 6 b 1+0.2+0.2 2 + ··· +0.2 7 c 1+ 1 6 + 1 6 2 + ··· + 1 6 10 d 1+1 1 +1 2 + ··· +1 10 E.7262 Determine, for each question, the exact value of the sum, then its value rounded to the hundredth : a 3 + 3 × 2 + 3 × 2 2 + · · · + 3 × 2 6 b 4 + 4 × 0.2 + 4 × 0.2 2 + · · · + 4 × 0.2 7 E.7004 Consider the geometric sequence with first term 1 and reason 2 . The sum of the first 13 terms of this sequence is : Which of the 3 answers below is correct? a 4095 b 8191 c 1 − 2 14 1 − 2 E.7046 Which of the following four state-ments is true? The sum S =1+2+2 2 +2 3 + ··· +2 30 is equal to : a − 1 + 2 31 b 1 − 2 31 c − 1 + 2 30 d 1 − 2 30 E.7267 Among the four statements below, only one is correct. Which statement? Justify your answer. The sequence u n is the geometric sequence with first term u 0 = 400 and reason 1 2 . The sum S = u 0 + u 1 + ··· + u 10 is equal to : a 2 × 1 − 0.5 10 b 2 × 1 − 0.5 11 c 800 × 1 − 0 ; 5 10 d 800 × 1 − 0.5 11 9. Threshold study E.7313 A commune opens a media library on 1 er January 2013 , the media library has a stock of 35 000 books from the old library, increased by 7 000 additional new books donated by the commune. Each year, the librarian is responsible for removing 5 % books that are too old or damaged. It is assumed that no new books are purchased by the library. We call u n the number, in thousands, of books available on 1 er January of the year (2013+ n ) . We give u 0 =42 . 1 a Justify that, for any natural number n , we have : u n +1 =0.95 × u n b Give the number of books in this library at 1 er January 2018 . 2 A natural language algorithm is proposed below. u ← 42 n ← 0 As long as u>10 u ← u × 0; 95 n ← n+1 End As long as a After running the algorithm, what does the value of the variable n represent. b Using your calculator, determine the value of the vari-able n after execution of the algorithm. E.7268 Among the four statements below, only one is correct. Which statement? Justify your answer. Consider the algorithm below : n ← 0 U ← 50 As long as U<120 U ← 1; 2 × U n ← n+1 End As long as At the end of execution, what is the value of the variable n : a 4 b 124.416 c 5 d 96 E.7206 A shopkeeper who has just opened a bou-tique notices that his sales started at 25 000 euros per month and increased every month by 2 % . He decides to model the progression of his sales by the follow-ing u n where u 0 represents the sales of the first month of opening. 1 Give the nature and characteristics of the sequence u n . 2 a Using the calculator, determine after how many months his sales will exceed 30 000 euros. b Complete the algorithm so that the value of the vari-able n has the value, at the end of its execution, of the number of months after opening so that its sales will exceed 30 000 euros ‘ .1 n ← 0 ‘ . 2 U ← 25 000 ‘ .3 As long as . .. make ‘ .4 n ← ... ‘ .5 U ← ... ‘ .6 End As long as 10. Sum of the terms of a geometric sequence https://chingmath.fr chapExoCorrec/7261 sacados/7261 chapExoCorrec/7262 sacados/7262 chapExoCorrec/7004 sacados/7004 chapExoCorrec/7046 sacados/7046 Extrait Antilles-Guyane Juin 2014 chapExoCorrec/7267 sacados/7267 Extrait Antilles-Guyannes Juin 2016 chapExoCorrec/7313 sacados/7313 chapExoCorrec/7268 sacados/7268 Extrait Antilles-Guyannes Juin 2016 chapExoCorrec/7206 sacados/7206
E.7263 On 1 er January 2017 , a sports as-sociation had 900 members. We observe that each month, 8 % of the association’s members do not renew their membership. 1 Determine the number of members on March 1 er 2017 . 2 We model the number of members n months after 1 er January 2017 thereafter u n . a Give the nature and characteristic elements of the se-quence u n . b Give the expression of the term u n according to its rank n . c Determine the number of members, rounded to the nearest whole number, on 1 er January 2018 . 3 Each member pays a monthly membership fee of 10 eu-ros. The association’s treasurer wants to calculate the total amount of membership fees for the year 2017 . The treasurer wants to use the following algorithm, in which the fourth line is incomplete (dotted line) . a Copy and complete the algorithm so that, at the end of its execution, the variable S has a value equal to the total amount of membership fees for the year 2017 . S ← 0 U ← 900 For N ranging from 1 to 12 S ← ... U ← 0.92 · U End For b What is the total amount of contributions received by the association during the year 2017 ? The amount is rounded to the nearest euro. E.7271 Consider the following algorithm: Function f(n) u ← 2 000 S ← 2 000 For i ranging from 2 to n u ← u × 1.008 S ← S+u End for We execute the function by providing it with the argument n value 5 . In the table below, summarize the values assigned to the vari-ables of the function f during its call: Value of i 2 Value of u 2 000 Value of S 2 000 E.7315 Without justification, give the value con-tained in the variable S after execution of this algorithm: 1 u ← 2 2 S ← 2 3 For i from 1 to 20 4 u ← u × 1; 05 5 S ← S+u . 6 End for E.7532 Consider the sequence u n , defined for any natural number n , geometric of first term 3 and reason 2 . Let S n be the sum of the ( n +1) first terms of the sequence u n . Establish the identity below for any natural number n : S n = 3 · 2 n +1 − 1 11. Geometric sequences, threshold and sum E.7314 A small town has a municipal bike rental service. The municipality wants to know how many bikes are in circulation and how much it costs. The manager of the bike rental service notes that between bikes that are unusable because they have been lost, stolen, or damaged, and new bikes that have been purchased, the number of usable bikes increases by 5 % each year. On January 1 er , 2017 , the fleet contains 200 usable bikes. We model the evolution of the number of usable bicycles by a sequence u n in which, for any natural number n , u n is the number of bicycles on January 1 er of year 2017+ n . Thus, u 0 =200 and, for any natural number n : u n +1 = 1.05 × u n . 1 a Justify the coefficient 1.05 in the expression of u n +1 as a function of u n . b How many bicycles will there be in this park on Jan-uary 1 er 2018 ? 2 The municipality has decided to stop purchasing new bi-cycles as soon as its stock exceeds 500 units. In what year will the municipal service’s stock exceed 500 bicycles for the first time? 3 To help maintain the rental service, the municipality has obtained a grant from the region that will be paid from 2017 to 2032 inclusive This subsidy amounts to 10 euros per bicycle available for rental. Assuming that the increase in the number of bikes re-mains constant at 5 % each year during this period, de-termine the total amount received from this subsidy from January 1 er to January 2017 to January 1 er to January 2032 . Give the exact value and the value rounded to two deci-mal places. https://chingmath.fr chapExoCorrec/7263 sacados/7263 chapExoCorrec/7271 sacados/7271 chapExoCorrec/7315 sacados/7315 chapExoCorrec/7532 sacados/7532 chapExoCorrec/7314 sacados/7314
S0Pournallantde0à24S×nFinPourAlgorithme1S0Pournallantde0à24S50×nFinPourAlgorithme2S50Pournallantde0à24S×nFinPourAlgorithme3 E.7264 Consider the geometric sequence u n , with common ratio 0.9 and first term u 0 =50 . 1 a Copy and complete the algorithm so that, at the end of its execution, the variable U has the value 25 e , which is the term of this sequence, i.e., u 24 : U ← ... For N ranging from 1 to 24 U ← ... End For b For any natural number n , express u n as a function of n . c Calculate u 24 and give an approximate value of the result to 10 − 3 near. 2 Using a calculator, give the smallest natural number n such that : u n < 0.01 . 3 We want to calculate the sum : S 24 = u 0 + u 1 + ··· + u 24 . Here are three proposed algorithms : a At the end of their execution, only one of these algo-rithms will have its variable S assigned the value of the sum u 24 . Specify which one and justify your answer. b Calculate the sum S 24 . Give an approximate value for the result, rounded to the nearest whole number. 12. Geometric sequences: limits E.7293 1 Consider the geometric sequence u n with first term 1 and common ratio 2 . a Consider the code : Function f(n) u ← 1 For i ranging from 1 to n u ← 2 × u End For Return u The function call f(n) returns the value of the term of the sequence u n of rank n to the program. Complete the table of values below : n 0 1 2 10 20 u n b Complete the dotted lines : lim n ↦→ + ∞ u n = : : : 2 a Modify the function f displays the terms of the se-quence v n defined by: v 0 = 2 ; v n +1 = 0.8 × v n b Complete the table of values rounded to the nearest thousandth : n 0 1 2 10 20 v n c Complete the dotted lines : lim n ↦→ + ∞ v n = : : : 3 Consider the geometric sequence w n with first term 50 and common ratio 0.8 . Let S n be the sum of the n +1 first terms of the sequence w n : S n = w 0 + w 1 + · · · + w n a Complete the dotted lines so that the function f(n) returns the term of rank n of the sequence S n : Function f(n) u ← 50 S ← 50 For i ranging from 1 to n u ← ... S ← S+... End for Return S b By making several calls to the function f , conjecture the limit of the sequence S n c Establish that : S n =250 · 1 − 0.8 n +1 d Justify the conjecture of the question b . E.7205 A website offers its subscribers movies to download. When it opens, 500 films are offered and each month the num-ber of films offered to subscribers increases by 6 % . We model the number of films offered by a geometric sequence u n where n denotes the number of months since the site opened. We therefore have : u 0 =500 . 1 Calculate u 1 and u 2 and give the result rounded to unity. 2 Express u n as a function of n . 3 Determine the limit of the sequence u n . E.7265 Consider the geometric sequence u n , of reason 0.9 and first term u 0 =50 . For any natural number n , note S n = u 0 + u 1 + ··· + u n . We admit that the sequence S n is increasing and that for any natural number n : S n =500 − 450 × 0.9 n . 1 Determine the limit of the sequence S n when n tends to + ∞ . 2 Alex assert that S n can exceed 500 for a sufficiently large value of the integer n . What do you think of his assertion? Justify the answer. https://chingmath.fr chapExoCorrec/7264 sacados/7264 S0Pournallantde0à24S×nFinPourAlgorithme1S0Pournallantde0à24S50×nFinPourAlgorithme2S50Pournallantde0à24S×nFinPourAlgorithme3 chapExoCorrec/7293 sacados/7293 chapExoCorrec/7205 sacados/7205 chapExoCorrec/7265 sacados/7265
E.7531 Consider the geometric sequence u n with first term u 0 =50 and reason 0.9 . 1 Determine the exact value of the term u 24 and give an approximate value of the result to within 10 − 3 . 2 For any natural number n , note : S n = u 0 + u 1 + ··· + u n a Determine the exact value of the sum S 24 . An approx-imate value of the result to the nearest unit will be given. We admit that the sequence S n is increasing and that for any natural number n : S n = 500 − 450 × 0.9 n b Determine the limit of the sequence S n when n tends to + ∞ . c Alex asserts that S n can exceed 500 for a sufficiently large value of the integer n . What do you think of his assertion? Justify the an-swer. 13. Geometric sequences: limits and thresholds E.7292 Consider the sequence u n defined by: u 0 = 0 ; u n +1 = 0.8 · u n + 0.1 for all n ∈ N ∗ 1 a Enter the algorithm below : u ← 0 For i ranging from 1 to n u ← 8 × u+ 0.1 End for b The variable n having value 10 , what do the different values of the variable u represent during the execution of the algorithm. c Running the algorithm step by step, what conjecture can be made about the values of the terms in the se-quence u n ? 2 a Enter the algorithm below : u ← 0 n ← 0 As long as u<0; 499 u ← 0.8 × u+0.1 n ← n+1 End as b At the end of the algorithm execution, what does the value of the variable n represent? E.7269 Note S n = u 1 + u 2 + ··· + u n the sum of the first n terms of a sequence u n , n being a non-zero natural number. We admit that : S n = − 250 000 + 250 000 × 1.008 n and that the sequence S n is increasing. 1 Determine the limit of the terms of the sequence S n when n tends to + ∞ . 2 Determine, using the calculator, from which rank on-wards, the sequence S n has a value greater than 125 000 . 14. Unclassified financial years E.6109 Consider the sequence u n de-fined by: u 0 = 115 ; u n +1 = 0.4 · u n + 120 for all n ∈ N 1 Consider the sequence v n defined for any natural num-ber n by: v n = u n − 200 . a Show that v n is a geometric sequence of reason 0.4 . Specify v 0 . b Express, for any natural number n , v n as a function of n . 2 Deduce that for any natural number n : u n = 200 − 85 × 0.4 n 3 Is the sequence u n convergent? Justify your answer. E.5557 Consider the sequence u n de-fined by: u 0 = 10 ; u n +1 = 0.9 u n + 1.2 for all n ∈ N . 1 Give the exact value of the first three terms of the se-quence u n . 2 Consider the sequence v n defined by: v n = u n − 12 for all n ∈ N . Show that the sequence v n is a geometric sequence of reason 0.9 and of first term − 2 . 3 a Express, for any natural number n , v n as a function of n . b Express, for any natural number n , u n as a function of n . c Deduce the limit of the sequence u n . https://chingmath.fr chapExoCorrec/7531 sacados/7531 chapExoCorrec/7292 sacados/7292 chapExoCorrec/7269 sacados/7269 chapExoCorrec/6109 sacados/6109 chapExoCorrec/5557 sacados/5557
E.5563 The production of Tahitian cul-tured pearls is an important economic activity for French Polynesia. To forecast the amounts realized from the export of Tahitian pearls, we model the situation by a sequence u n . We note u 0 the amount in 2011 , in millions of euros, and u n the amount in 2011+ n , in millions of euros. The following information is given : In 2011 , the gross value of pearl products was 63 million euros. It is assumed that the amount of exports made fell every year by 8 % . 1 Show that the sequence u n is a geometric sequence whose reason will be specified. 2 Express, for any natural number n , u n as a function of n . 3 With this model, what amount can we forecast for the export of pearl products from French Polynesia in 2016 ? Round the result to the nearest million euros. E.5560 Consider the sequence u n de-fined by: u 0 = 42 ; u n +1 = 0.95 · u n + 4 for everything n ∈ N 1 Determine the exact value of the first three terms of the sequence u n 2 Consider the sequence w n defined by: v n = u n − 80 for all n ∈ N . Show that v n is a geometric sequence of reason q =0.95 and of first term − 38 . 3 a Express, for any natural number n , v n as a function of n . b Express, for any natural number n , u n as a function of n . c Determine the limit of the sequence u n . E.6095 Consider the sequence a n defined by: a 0 = 8 ; a n +1 = 3 4 · a n + 4 for all n ∈ N 1 For any natural number n , note : u n = a n − 16 . a Show that the sequence u n is a geometric sequence of reason 3 4 . b Express u n in terms of n . 2 Express a n as a function of n . What can we say about the convergence of the sequence a n ? E.3578 Two companies, Ultra-eau ( U ) and Vital-eau ( V ) , share the market for bottled water coolers in businesses in a large city. In 2013 , company U had 45 % of the market and company V had the rest. Each year, company U retains 90 % of its customers, while the others choose company V . As for the company V , it retains 85 % of its customers, while the others choose the company U . A customer is chosen at random each year and the following is noted for each natural number n : u n the probability that they will be a customer of the company U in the year 2013+ n , thus u 0 =0.45 ; v n the probability that he is a customer of the company V in the year 2013+ n . 1 Represent the situation using a probabilistic graph with vertices U and V . 2 Give v 0 , calculate u 1 et v 1 . 3 Consider the function f (incomplete) , extracted from an algorithm, given below. Called with a non-zero natural integer n as argument, this returns the pair of values ( u n ; v n ) . Complete the lines ( ‘ .3 ) and ( ‘ .6 ) of the function to ob-tain the expected result. ‘ .1 Fonction f(N) ‘ .2 U ← 0.45 ‘ .3 V ← ... ‘ . 4 For i allowing from 1 to N ‘ .5 U ← 0.9 × U+0.15 × V ‘ . 6 V ← ... ‘ .7 Finish For ‘ .8 Send ( U ; V) 4 We admit that, for any natural number n : u n +1 = 0.75 · u n + 0.15 . We note, for any natural number n : w n = u n − 0.6 a Show that the sequence w n est a geometric sequence of reason 0.75 . b What is the limit of the sequence w n ? Deduce the limit of the sequence u n . Interpret the result in the context of this exercise. https://chingmath.fr chapExoCorrec/5563 sacados/5563 chapExoCorrec/5560 sacados/5560 chapExoCorrec/6095 sacados/6095 chapExoCorrec/3578 sacados/3578