Grade 11 - STMG / Sequence 47 exercises (including 46 corrected)

a
1erétape2eétape3eétape 1erétape2ièmeétape3ièmeétape 1234567ABCDTempsPopulationdelasoucheAPopulationdelasoucheBPopulationtotale0200300 a0×:::a1×:::a2×:::a3×:::a4×:::×:::×:::×::: b0−:::b1−:::b2−:::b3−:::b4−:::−:::−:::− 1. Introduction E.11420 Here are some examples of number sequences : a ( 2 ; 5 ; 8 ; 11 ; 14 ; : : : ) b ( 2 ; 6 ; 18 ; 54 ; 162 ; : : : ) c ( 6 ; − 6 ; 6 ; − 6 ; 6 ; : : : ) d ( 1 ; 3 ; 7 ; 15 ; 31 ; : : : ) Determine the next three terms in each of these sequences. E.7387 The figures below are constructed using small wooden sticks. For n a strictly positive integer ( n ∈ N ∗ ) , we note u n the num-ber of sticks needed for construction in step n . Give the values of the first four terms of the sequence u n . E.7323 Consider the construction of a house of cards : For n a strictly positive integer ( n ∈ N ∗ ) , note u n the number of cards needed to build the house of cards at the n ième step. Give the first four terms of the sequence u n E.7320 1 Consider the sequence of numbers : 1 ; 1 2 ; 1 3 ; 1 4 ; 1 5 ; : : : a Determine the next three terms in this sequence. b Consider the function f defined by the relation: f ( x ) = 1 x . What is the relationship between the function f and the sequence of numbers? 2 a Consider the sequence of numbers : 1 ; 2 ; 3 ; 2 ; 5 ; 6 : : : Which function g is this sequence of numbers related to? b Consider the sequence of numbers : 1 2 ; 2 3 ; 3 4 ; 4 5 ; 5 6 ; 6 7 ; : : : Which function h is this sequence of numbers related to? E.11419 Scientists are studying a bacterial culture 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 a At time ˇ 0 min ı, what percentage of the total bacte-ria are represented by bacteria of strain A ? b At time ˇ 1 min ı, , what percentage of the total bacteria are represented by the bacteria of strain A ? c Complete the table below : 2 n denotes a natural number ( n ∈ R ) . Let a n be the population of bacteria of strain A at time ˇ n min ı ; thus, a 0 =200 . We note b n the bacterial population of strain B at time ˇ n min ı ; as follows b 0 = 300 . Complete the dotted lines below : a 1 = a 0 . . . . . . . . . . . . b 1 = b 0 . . . . . . . . . . . . a 2 = a 1 . . . . . . . . . . . . b 2 = b 1 . . . . . . . . . . . . a 3 = a 2 . . . . . . . . . . . . b 3 = b 2 . . . . . . . . . . . . a 4 = a 3 . . . . . . . . . . . . b 4 = b 3 . . . . . . . . . . . . We generalize by a n +1 = a n . . . . . . . . . . . . b n +1 = b n . . . . . . . . . . . . 3 Complete the two diagrams below : a b https://chingmath.fr chapExoCorrec/11420 sacados/11420 chapExoCorrec/7387 sacados/7387 1erétape2eétape3eétape chapExoCorrec/7323 sacados/7323 1erétape2ièmeétape3ièmeétape chapExoCorrec/7320 sacados/7320 chapExoCorrec/11419 sacados/11419 1234567ABCDTempsPopulationdelasoucheAPopulationdelasoucheBPopulationtotale0200300 a0×:::a1×:::a2×:::a3×:::a4×:::×:::×:::×::: b0−:::b1−:::b2−:::b3−:::b4−:::−:::−:::−
3u0×2u1×2u2×2u3×2u4 −2v0v1v2v3v4 u0u1u2u3 4 Complete the dotted lines : a 1 = a 0 . . . . . . . . . . . . b 1 = b 0 . . . . . . . . . . . . a 2 = a 0 . . . . . . . . . . . . b 2 = b 0 . . . . . . . . . . . . a 3 = a 0 . . . . . . . . . . . . b 3 = b 0 . . . . . . . . . . . . a 4 = a 0 . . . . . . . . . . . . b 4 = b 0 . . . . . . . . . . . . We generalize by: a n = a 0 . . . . . . . . . . . . b n = b 0 . . . . . . . . . . . . 2. Introduction: Definition by Recursion E.7324 In this exercise, the sequences are defined for a value of n integer positive or zero ( n ∈ N ) : 1 Consider the sequence u n whose first term u 0 has the value 3 and whose one term is passed on to the next by multiplying by the same number. Complete the diagram below to find the first terms of the sequence u n : 2 Consider the sequence v n whose first term u 0 has the value − 2 and whose next term is passed by adding by the same number. Complete the diagram below to find the first terms of the sequence v n : E.7340 1 Consider the following sequence of numbers : 2 ; 3 ; 5 ; 8 ; 12 ; 17 ; 23 ; 30 a In this sequence, which term succeeds 12 ? b In this sequence, what is the term that precedes 8 ? 2 We consider a sequence of numbers that we note u and whose terms we index using a positive or zero natural number: thus, we ˇ number ı the values of the sequence starting with 0 : u 0 ; u 1 ; u 2 ; u 3 ; · · · ; u n − 1 ; u n ; u n +1 a What is the successor term to u 2 ? b What is the predecessor term of u 4 ? c What is the successor term of u n ? d What is the successor term to u n +2 ? e What is the predecessor term of u n ? f What is the predecessor term of u n +2 ? E.7322 In this exercise, the sequences are indexed using a positive or zero integer n ( n ∈ N ) : 1 Consider the sequence whose first term is 2 and whose successor ˇ is twice its prédécesseur ı Construct the first four terms of the sequence. 2 Consider the sequence whose first term is worth − 3 and whose ˇ successor is worth its predecessor increased by 3 . ı Construct the first four terms of the sequence. 3 Consider the sequence whose terms are indexed from 0 and whose ˇ the value of a term is the square of its rang ı. 3. Arithmetic sequences: first terms E.7338 In this exercise, the sequences are defined for integers n positive or zero: 1 Consider the sequence u n arithmetic with first term 3 and reason 5 . Complete the diagram below to obtain the first four terms of the sequence : 2 Consider the sequence v n arithmetic with first term 6 and reason − 2 . Complete the dotted lines below to obtain the first four terms of the sequence : v 0 = : : : : : : v 1 = : : : : : : + ( − 2) = : : : : : : v 2 = : : : : : : + ( − 2) = : : : : : : v 3 = : : : : : : + ( − 2) = : : : : : : https://chingmath.fr chapExoCorrec/7324 sacados/7324 3u0×2u1×2u2×2u3×2u4 −2v0v1v2v3v4 chapExoCorrec/7340 sacados/7340 chapExoCorrec/7322 sacados/7322 chapExoCorrec/7338 sacados/7338 u0u1u2u3
u0u1u2u3 u0u1u2u3 v0v1v2v3 E.7341 In this exercise, the terms of the sequences have rank as an integer n positive or zero ( n ∈ N ) . 1 Determine the first five terms of the suite u n arithmetic of first term 2 and reason 3 . 2 Determine the first five terms of the sequence v n arith-metic with first term 3 and reason − 3 2 . E.4572 Consider the sequence u n , where n ∈ N , arithmetic with first term 3 and reason 5 . Determine the first five terms of this sequence. E.11549 Consider the sequence u n , where n ∈ N , arithmetic with first term 11 and common difference − 2 . Determine the first four terms of this sequence. 4. Geometric sequences: first terms E.7339 In this exercise, the sequences are indexed using an integer n positive or zero ( n ∈ N ) . 1 Consider the sequence u n geometric with first term 2 and reason 3 . Complete the diagram below to obtain the first four terms of the sequence : 2 Consider the sequence v n geometrically defined by: v 0 = − 2 ; v n +1 = 1 2 · v n Complete the dotted lines below to obtain the first four terms of the sequence : v 0 = : : : : : : v 1 = : : : : : : × 1 2 = : : : : : : v 2 = : : : : : : × 1 2 = : : : : : : v 3 = : : : : : : × 1 2 = : : : : : : E.7342 In this exercise, the terms of the sequences have rank as an integer n positive or zero ( n ∈ N ) . 1 Determine the first four terms of the suite u n geometric with first term 2 and reason 3 . 2 Determine the first four terms of the sequence v n geo-metric with first term 3 and reason − 3 2 . E.9495 Consider the sequence v n geomet-ric defined, for any n ∈ N , by: v 0 = − 2 ; v n +1 = 1 2 · v n Determine the value of the first 6 terms of the sequence v n . E.11550 Consider the geometric sequence v n defined, for all n ∈ N , by: v 0 = 64 ; v n +1 = 0 ; 2 · v n Determine the value of the first 4 terms of the sequence v n . 5. Arithmetic and geometric sequences: first terms E.11548 In this exercise, sequences are de-fined for a positive integer or zero value of n ( n ∈ N ) : 1 Consider the arithmetic sequence u n with first term 3 and common ratio 2 . Complete the diagram to find the first terms of the se-quence u n : 2 Consider the geometric sequence v n with first term 0 ; 375 and common ratio 4 . Complete the diagram below to find the first terms of the sequence v n : E.7356 In this exercise, the terms of the sequences have rank as an integer n positive or zero ( n ∈ N ) . 1 Determine the first five terms of the suite u n arithmetic with first term 10 and reason − 3 . 2 Determine the first five terms of the sequence v n geo-metric of first term 3 and reason. 1 2 . If necessary, values rounded to the nearest hundredth of the terms of the sequence v n . E.7388 In this exercise, the terms of the sequences have rank as an integer n positive or zero ( n ∈ N ) . 1 Determine the first four terms of the suite u n arith-metic with first term 3 and reason − 3 4 . 2 Determine the first four terms of the sequence v n geo-metric of first term 5 and reason 1 4 . If necessary, we will give the values, rounded to the near-est hundredth, of the terms in the sequence v n . 6. Non-arithmetic and non-geometric sequences: https://chingmath.fr chapExoCorrec/7341 sacados/7341 chapExoCorrec/4572 sacados/4572 chapExoCorrec/11549 sacados/11549 chapExoCorrec/7339 sacados/7339 u0u1u2u3 chapExoCorrec/7342 sacados/7342 chapExoCorrec/9495 sacados/9495 chapExoCorrec/11550 sacados/11550 chapExoCorrec/11548 sacados/11548 u0u1u2u3 v0v1v2v3 chapExoCorrec/7356 sacados/7356 chapExoCorrec/7388 sacados/7388
E.7337 In this exercise, the sequences are indexed using a positive or zero integer n ( n ∈ N ) : 1 Consider the sequence u 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 se-quence. 2 Consider the sequence v n 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 se-quence. E.7390 In this exercise, the sequences are indexed using a positive or zero integer n ( n ∈ N ) : 1 Consider the sequence u n whose first terms are: u 0 = 5 ; u 1 = 6.4 ; u 2 = 7.8 ; u 3 = 9 Justify that the sequence u n is not an arithmetic se- quence. 2 Consider the sequence v n whose first terms are: v 0 = 20 ; v 1 = 8 ; v 2 = 3.2 ; v 3 = 1.44 Justify that the sequence v n is not a geometric se-quence. E.11551 In this exercise, sequences are in-dexed using a positive integer n or zero ( n ∈ N ) : 1 Consider the sequence u n whose first terms are: u 0 = 5 ; u 1 = 6 ; 35 ; u 2 = 7 ; 7 ; u 3 = 9 ; 15 ; u 4 = 10 ; 5 Justify that the sequence u n is not an arithmetic se-quence. 2 Consider the sequence v n whose first terms are: v 0 = 4000 ; v 1 =2560 ; v 2 =1600 ; v 3 =1000 ; v 4 =625 Justify that the sequence v n is not a geometric se-quence. 7. Arithmetic sequences: explicit formula E.11452 Consider the sequence u n defined for n ∈ N and arithmetic with first term 5 and common differ-ence 2 . 1 Let n be a positive integer. ( n ∈ N ) , Give the expression for the term u n of rank n . 2 Give the value of the term of rank 100 . E.7346 Consider the sequence u 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 . E.4575 Consider the sequence u n defined explicitly by the relation 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. E.7568 Consider the sequence u n , where n ∈ N , defined by the recurrence relation: u 0 = 5 ; u n +1 = u n − 2 1 What is the nature of this sequence? 2 Give the explicit formula giving the value of u n as a func-tion of n . 3 Determine the value of u 20 . E.11552 Consider the sequence u n , where n ∈ N , defined by the recurrence relation: u 0 = − 4 ; u n +1 = u n + 2 1 What is the nature of this sequence? 2 Give the explicit formula giving the value of u n as a func-tion of n . 3 Determine the value of u 20 . 8. Geometric sequences: explicit formula E.11453 Let u n be the sequence defined for n ∈ N and geometric with first term 10 and common ratio 0 ; 9 . 1 Give the explicit formula giving the value of the term u n of rank n (where n ∈ N ) . 2 Determine the value of the term of rank 10 rounded to the nearest thousandth. E.7865 Consider the sequence v n n ∈ N de-fined by the recurrence relation: v 0 = 64 ; v n +1 = 1 2 · v n 1 What is the nature of this sequence? 2 Give the explicit formula giving the value of v n as a func-tion of n . 3 Determine the value of v 6 . https://chingmath.fr chapExoCorrec/7337 sacados/7337 chapExoCorrec/7390 sacados/7390 chapExoCorrec/11551 sacados/11551 chapExoCorrec/11452 sacados/11452 chapExoCorrec/7346 sacados/7346 chapExoCorrec/4575 sacados/4575 chapExoCorrec/7568 sacados/7568 chapExoCorrec/11552 sacados/11552 chapExoCorrec/11453 sacados/11453 chapExoCorrec/7865 sacados/7865
E.11553 Consider the sequence v n defined by the recurrence relation: v 0 = 2 ; 048 ; v n +1 = 1 ; 25 · v n for all n ∈ N 1 Give the nature of the sequence v n and its characteris-tic elements. 2 Give the explicit formula giving the value of v n as a func-tion of n . 3 Determine the value of v 5 . E.9486 Consider the sequence v n defined by the recurrence relation: v 0 = 64 ; v n +1 = 1 2 · v n for all n ∈ N 1 Give the nature of the suite v n and its characteristic elements. 2 Give the explicit formula giving the value of v n as a func-tion of n . 3 Determine the value of v 6 . 9. Arithmetic sequences: determining the rank E.11480 Consider the sequence u n defined for all n ∈ N and arithmetic with first term 4 and common dif-ference 3 . 1 Give the explicit formula for the sequence u n . 2 Determine the rank n such that : u n = 115 E.11482 Consider the sequence u n defined for all n ∈ N and arithmetic with first term 9 and common dif-ference − 2 . 1 Give the explicit formula for the sequence u n . 2 Determine the rank n such that : u n = − 137 E.11484 Consider the sequence u n defined for all n ∈ N and arithmetic with first term 5 and common dif-ference 0 ; 3 . 1 Give the explicit formula for the sequence u n . 2 Determine the rank n such that : u n = 30 ; 2 E.11554 Consider the sequence u n defined for all n ∈ N and arithmetic with first term 4 and common dif-ference 0 ; 6 . 1 Give the explicit formula for the sequence u n . 2 Determine the rank n such that : u n = 37 10. Geometric sequences: determining the rank E.11481 Consider the geometric sequence u n defined for all n ∈ N with first term 9 and common ratio 2 . 1 Give the explicit formula for the sequence u n . 2 Determine the rank n such that : u n = 2 359 296 2 2 = 4 ; 2 3 = 8 ; 2 4 = 16 ; 2 5 = 32 2 6 = 64 ; 2 7 = 128 ; 2 8 = 256 ; 2 9 = 512 2 10 = 1024 ; 2 11 = 2048 ; 2 12 = 4096 2 13 = 8192 ; 2 14 = 16384 ; 2 15 = 32768 2 16 = 65536 ; 2 17 = 131072 ; 2 18 = 262144 2 19 = 524288 ; 2 20 = 1048576 ; 2 21 = 2097152 E.11483 Consider the geometric sequence u n defined for all n ∈ N with first term 5 and common ratio 3 . 1 Give the explicit formula for the sequence u n . 2 Determine the rank n such that : u n = 17 433 922 005 3 2 = 9 ; 3 3 = 27 ; 3 4 = 81 ; 3 5 = 243 3 6 = 729 ; 3 7 = 2187 ; 3 8 = 6561 ; 3 9 = 19683 3 10 = 59049 ; 3 11 = 177147 ; 3 12 = 531441 3 13 = 1594323 ; 3 14 = 4782969 ; 3 15 = 14348907 3 16 = 43046721 ; 3 17 = 129140163 ; 3 18 = 387420489 3 19 = 1162261467 ; 3 20 = 3486784401 3 21 = 10460353203 ; 3 22 = 31381059609 E.11555 Consider the geometric sequence u n defined for all n ∈ N with first term 23 and common ratio 2 . 1 Give the explicit formula for the sequence u n . 2 Determine the rank n such that : u n = 188 416 2 2 = 4 ; 2 3 = 8 ; 2 4 = 16 ; 2 5 = 32 2 6 = 64 ; 2 7 = 128 ; 2 8 = 256 ; 2 9 = 512 2 10 = 1024 ; 2 11 = 2048 ; 2 12 = 4096 2 13 = 8192 ; 2 14 = 16384 ; 2 15 = 32768 2 16 = 65536 ; 2 17 = 131072 ; 2 18 = 262144 2 19 = 524288 ; 2 20 = 1048576 ; 2 21 = 2097152 11. Arithmetic and geometric sequences: characteristic elements https://chingmath.fr chapExoCorrec/11553 sacados/11553 chapExoCorrec/9486 sacados/9486 chapExoCorrec/11480 sacados/11480 chapExoCorrec/11482 sacados/11482 chapExoCorrec/11484 sacados/11484 chapExoCorrec/11554 sacados/11554 chapExoCorrec/11481 sacados/11481 chapExoCorrec/11483 sacados/11483 chapExoCorrec/11555 sacados/11555
E.7349 1 Consider the arithmetic sequence u n defined for any positive integer n or zero ( n ∈ N ) of which only the follow-ing two terms are known : u 4 = 3 ; u 7 = 15 Determine the first term and the common difference of this arithmetic sequence. 2 Consider the geometric sequence v n defined for any positive integer n or zero ( n ∈ N ) of which only the fol-lowing two terms are known : v 2 = 2 ; v 5 = 54 Determine the first term and the common ratio of this geometric sequence. Hint: Here are some numerical values to know : 2 2 = 4 2 3 = 8 2 4 = 16 2 5 = 32 3 2 = 9 3 3 = 27 3 4 = 81 3 5 = 243 4 2 = 16 4 3 = 64 4 4 = 256 4 5 = 1024 5 2 = 25 5 3 = 125 5 4 = 625 5 5 = 3125 6 2 = 36 6 3 = 216 6 4 = 1296 6 5 = 7776 E.7357 1 Consider the arithmetic sequence u n defined for any positive integer n or zero ( n ∈ N ) of which only the follow-ing two terms are known : u 3 = 4 ; 5 ; u 6 = 9 Determine the first term and the common difference of this arithmetic sequence. 2 Consider the geometric sequence v n defined for any positive integer n or zero ( n ∈ N ) of which only the fol-lowing two terms are known : v 2 = 8 ; v 5 = 1 Determine the first term and the common ratio of this arithmetic sequence. Hint: Here are some numerical values to know : 0 ; 1 2 = 0 ; 1 0 ; 1 3 = 0 ; 01 0 ; 1 4 = 0 ; 001 0 ; 2 2 = 0 ; 04 0 ; 2 3 = 0 ; 008 0 ; 2 4 = 0 ; 0016 0 ; 25 2 = 0 ; 0625 0 ; 25 3 = 0 ; 15625 0 ; 25 4 = 0 ; 00390625 0 ; 4 2 = 0 ; 16 0 ; 4 3 = 0 ; 064 0 ; 4 4 = 0 ; 0256 0 ; 5 2 = 0 ; 25 0 ; 5 3 = 0 ; 125 0 ; 5 4 = 0 ; 0625 0 ; 6 2 = 0 ; 36 0 ; 6 3 = 0 ; 216 0 ; 6 4 = 0 ; 1296 0 ; 75 2 = 0 ; 5625 0 ; 75 3 = 0 ; 421875 0 ; 75 4 = 0 ; 31640625 0 ; 8 2 = 0 ; 64 0 ; 8 3 = 0 ; 512 0 ; 8 4 = 0 ; 4096 E.7389 1 Consider the arithmetic sequence u n defined for any positive integer n or zero ( n ∈ N ) of which only the follow-ing two terms are known : u 6 = 5 ; u 15 = 15 ; 8 Determine the first term and the common difference of this arithmetic sequence. 2 Consider the geometric sequence v n defined for any positive integer n or zero ( n ∈ N ) of which only the fol-lowing two terms are known : v 2 = 32 ; v 5 = 131 ; 072 Determine the first term and the common ratio of this geometric sequence. Hint: Here are some numerical values to know : 1 ; 2 2 = 1 ; 44 1 ; 2 3 = 1 ; 728 1 ; 2 4 = 2 ; 0736 1 ; 25 2 = 1 ; 5625 1 ; 25 3 = 1 ; 953125 1 ; 25 4 = 2 ; 44140625 1 ; 4 2 = 1 ; 96 1 ; 4 3 = 2 ; 744 1 ; 4 4 = 3 ; 8416 1 ; 5 2 = 2 ; 25 1 ; 5 3 = 3 ; 375 1 ; 5 4 = 5 ; 0625 1 ; 6 2 = 2 ; 56 1 ; 6 3 = 4 ; 096 1 ; 6 4 = 6 ; 5536 1 ; 75 2 = 3 ; 0625 1 ; 75 3 = 5 ; 359375 1 ; 75 4 = 9 ; 37890625 1 ; 8 2 = 3 ; 24 1 ; 8 3 = 5 ; 832 1 ; 8 4 = 10 ; 4976 12. Further developments E.7347 Mandine hired Arthur on January 1 st , 2009, with a salary of 1525 e and offered him two types of advancement : Every January 1 st , his salary will increase by 32 e . Every January 1 st , his salary will increase 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 ? https://chingmath.fr chapExoCorrec/7349 sacados/7349 chapExoCorrec/7357 sacados/7357 chapExoCorrec/7389 sacados/7389 chapExoCorrec/7347 sacados/7347
1234567ABCDAnnéePopulationdePau1janvierPopulationdeVau1janvierPopulationdeIau1janvier2002200000300000 p0×:::p1×:::p2×:::p3×:::p4×:::×:::×:::×::: v0−:::v1−:::v2−:::v3−:::v4−:::−:::−:::−::: E.7348 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 natural number ( n ∈ R ) . Let p n be the population of P on January 1 er (2002+ n ) ; thus : p 0 =200 000 . We denote v n as the population of V on January 1 er (2002+ n ) ; thus : 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 E.7350 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 . 13. Share E.11621 Victor sort un plat du four. La température du plat est alors égale à 180 ° C. Il place ce plat dans une pièce dont la température est égale à 25 ° C. Le plat refroidit. Le plat ne pourra être servi que lorsque sa tempéra-ture sera devenue inférieure ou égale à 40 ° C. On étudie le refroidissement du plat selon deux modèles math-ématiques. Partie A : Premier modèle On suppose que la baisse de la température du plat est propor-tionnelle à la durée du refroidissement, c’est-à-dire au nombre de minutes écoulées depuis la sortie du four. On constate que 3 minutes après la sortie du four, la tempéra-ture du plat est égale à 105 ° C. 1 De combien de degrés le plat a-t-il baissé en 3 minutes ? En 1 minute ? 2 Vérifier que la température du plat, 5 minutes après la sortie du four, est égale à 55 ° C. 3 Selon ce modèle, quelle serait la température du plat, 8 minutes après la sortie du four ? Ce premier modèle semble-t-il pertinent ? Partie B: Second modèle On dispose toujours des données suivantes : la température de la pièce est égale à 25 ° C ; la température du plat à la sortie du four est égale à 180 ° C ; la température du plat, 3 minutes après la sortie du four, est égale à 105 ° C. Pour tout entier naturel n , on note U n la différence entre la température du plat et la température de la pièce, n minutes après la sortie du four. Exemple: 3 minutes après la sortie du four, l’écart avec la température de la pièce est égal à 105 − 25 = 80 . On a donc U 3 = 80 . 1 Justifier que U 0 = 155 . 2 On suppose que chaque minute la différence U n diminue de 20%. a Justifier que, pour tout entier naturel n , on a U n +1 = 0 ; 8 U n . b En déduire la nature de la suite ( U n ) et donner sa rai-son. c Exprimer U n en fonction de n , pour tout entier naturel n . d On dispose des données suivantes : n 3 4 5 6 7 8 9 10 11 12 13 14 15 U n (arrondi à 10 − 1 ) 80 64 51 , 2 41 32 , 8 26 , 2 21 16 , 8 13 , 4 10 , 7 8 , 6 6 , 9 5 , 5 Au bout de combien de minutes Victor pourra-t-il servir le plat ? https://chingmath.fr chapExoCorrec/7348 sacados/7348 1234567ABCDAnnéePopulationdePau1janvierPopulationdeVau1janvierPopulationdeIau1janvier2002200000300000 p0×:::p1×:::p2×:::p3×:::p4×:::×:::×:::×::: v0−:::v1−:::v2−:::v3−:::v4−:::−:::−:::−::: chapExoCorrec/7350 sacados/7350 sacados/11621
14. Unclassified financial years E.7325 Consider the numerical sequences whose terms are defined for any strictly positive integer n ( n ∈ N ∗ ) by the relations below : a u n = 2 n b v n = 3 n − 4 c w n = n 2 + 3 d x n = 2 n Determine the first five terms of each of these sequences. E.7321 For each question, the sequence is de-fined for strictly positive values of n ( n ∈ N ∗ ) . Determine the first four terms of the sequence u n n ∈ N : a u n = n + 1 n + 2 b u n = n 2 + n + 1 https://chingmath.fr chapExoCorrec/7325 sacados/7325 chapExoCorrec/7321 sacados/7321