Grade 11 / Arithmetic, geometric and other sequences 94 exercises corrected

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ChingQuizz : 18 exercises available for Quizz assessment : 1. A few reminders E.2905 Reminders: Consider a value x undergoing a change to obtain the value y : Reduction of a % : y = x · 1 − a 100 Increase of a % : y = x · 1+ a 100 1 Find the multiplier coefficients representing each of the following changes : a +10 % b +2 ; 5 % c +115 % d − 22 % e − 10 ; 7 % f − 65 % 2 For each multiplier coefficient, find the associated change and the corresponding percentage: a 1 ; 02 b 1 ; 375 c 2 ; 1 d 0 ; 15 e 0 ; 85 f 0 ; 912 2. Introduction to suites E.2379 1 Consider the logical sequence : 3 ; 7 ; 11 ; 15 ; 19 Noting successively u 0 , u 1 , u 2 , . . . the terms of this se-quence, give the value of u 6 . Hint: i.e. the term 3 is noted u 0 , the term 4 is noted u 1 , the term 11 is noted u 2 . . . 2 Consider the logical sequence : 2 ; 4 ; 6 ; 8 ; 10 Noting v 0 , v 1 , v 2 , . . . the terms of this sequence, give the value of v 5 . 3 Consider the logical sequence : -1 ; 2 ; − 4 ; 8 ; − 16 ; 32 Noting w 0 , w 1 , . . . the terms of this sequence, give the value of w 6 . 4 Consider the logical sequence : 1 ; 3 ; 17 ; 5 ; 33 ; 41 Noting s 0 , s 1 , . . . the terms of this sequence, give the value of s 6 . 5 Consider the logical sequence : 2 ; 3 2 ; 4 3 ; 5 4 Noting t 0 , t 1 , . . . the terms of this sequence, give the value t 5 E.6517 Consider the following two processes for obtaining numbers : Procedure A The given number is multi-plied by 3 Procedure B To the given number, we subtract 2 . For each question, give the first six terms obtained, repeating the instructions as many times as necessary. 1 The starting number is 3 and we repeat the procedure A ; 2 The starting number is 11 and we repeat the procedure B . E.7308 Consider the two algorithms below : Algorithme 1 u ← 4 For i ranging from 1 to 53 u ← u + 3 End For Algorithme 2 u ← 1 For i ranging from 1 to 4 u ← 2 × u + 1 End For For each of the algorithms, give the value contained in the variable u after execution of the algorithm. E.6516 Complete the logical sequences of numbers to obtain the first 8 terms of each : a 4 - 7 - 10 - 13 - . . . b 3 - 6 - 12 - 24 - . . . c 20 - 19 - 17 - 14 - . . . d 5 - 7 - 11 - 17 - . . . e 1 - 4 - 9 - 16 - . . . https://chingmath.fr quizzChap/752 chapExoCorrec/2905 sacados/2905 chapExoCorrec/2379 sacados/2379 chapExoCorrec/6517 sacados/6517 chapExoCorrec/7308 sacados/7308 chapExoCorrec/6516 sacados/6516
E.7183 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 ; : : : ) For each of these sequences of numbers ; find the relationship that gives a value as a function of the previous values. E.9483 Consider the logical sequences of numbers below : a ( 0 ; 2 ; 4 ; 6 ; 8 ; : : : ) b ( 1 ; 6 ; 11 ; 16 ; 21 ; : : : ) c ( 1 ; 2 ; 4 ; 8 ; 16 ; 32 ; : : : ) d ( 1 ; 2 ; 3 ; 2 ; 5 ; 6 ; : : : ) e ( 1 ; 1 2 ; 1 3 ; 1 4 ; 1 5 ; : : : ) Noting u 0 , u 1 , u 2 , . . . the successive terms of this sequence, give the value of the term u 6 of each of these sequences. 3. Introduction to vocabulary E.7194 1 Consider the sequence of numbers below : 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 Generally speaking, we indicate the terms of a sequence by using the position of the term in the sequence as an index (we start the indexation at 0 ) : u 0 ; u 1 ; u 2 ; u 3 ; · · · ; u n − 1 ; u n ; u n +1 a What is the successor term of 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.7192 1 Consider the sequence whose first term is 2 and whose successor ˇ has a value twice that of its prédécesseur ı Construct the first four terms of the sequence. 2 Consider the sequence whose first term is − 3 and whose successor ˇ has the value of 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 value of a term is the square of its rang ı. Construct the first four terms of the sequence. E.6519 Consider the three sequences whose first term is 2 and defined as follows : 1 a term is worth the inverse of its predecessor to which we add 2 2 a term is worth double its predecessor to which we add 2 3 a term is worth double its rank to which we add 2 . Associate the corresponding definition with each of these se-quences : a u n =2 · n +2 b u n +1 = 1 u n +2 c u n +1 =2 · u n +2 4. Introduction to arithmetic and geometric sequence generation E. 2906 Scientists are studying a culture of bacteria containing two strains to be named A and B . At the start of the experiment (at time ˇ0ı) , there are 200 bacteria of strains A and 300 bacteria of strains B . The scientists note the following developments : every minute, the population of A bacteria increases by 10 % , while that of the B strain decreases by 20 bacteria. 1 a At time ˇ 0 min ı, what percentage is represented by bacteria of strain A relative to all bacteria? b At time ˇ 1 min ı, what percentage is represented by bacteria of strain A relative to all bacteria? c Complete the table below : 2 n denotes a natural integer ( n ∈ R ) . https://chingmath.fr chapExoCorrec/7183 sacados/7183 chapExoCorrec/9483 sacados/9483 chapExoCorrec/7194 sacados/7194 chapExoCorrec/7192 sacados/7192 chapExoCorrec/6519 sacados/6519 chapExoCorrec/2906 sacados/2906 1234567ABCDTempsPopulationdelasoucheAPopulationdelasoucheBPopulationtotale0200300
We denote a n the population of bacteria of strain A at time ˇ n min ı ; thus, a 0 =200 . Note b n the bacterial population of strain B at time ˇ n min ı ; thus b 0 = 300 . Complete the blanks 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 4 Complete the blanks : 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 . . . . . . . . . . . . E.2372 The Mandine company hires Arthur on 1 er January 2009 with a salary of 1525 e and offers him two types of advancement : Every 1 er January, his salary will be increased by 32 e . Each 1 er January, his salary increases by 2 % . 1 Complete the following table, rounding values to the nearest tenth : Année 2009 2010 2011 2012 Avancement A Avancement B Année 2013 2014 2015 2016 Avancement A Avancement B 2 Starting in which year will Arthur have a higher salary by choosing advancement B ? E.7195 A job seeker is offered two offers : An initial salary of 1150 euros per month and an increase of 5 % per month. We note a n the sequence of these monthly earnings with this proposal. An initial salary of 1200 euros per month and an increase of 3 % per month. We note b n the sequence of these monthly earnings 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, 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 month, which proposal makes the salary more advantageous? b Looking at the sum received at the end of the five months, which proposal is the most advantageous? 5. Arithmetic sequences: first terms E.8523 Definition: We call arithmetic sequence any sequence of numbers whose successor of a term is obtained by adding to the one always the same number. This number is called the reason of this arithmetic se-quence. Consider the sequence u n arithmetic, where n ∈ N , of first term u 0 =5 and reason 3 . Copy and complete the line below to obtain the first five terms of this sequence : u 0 = : : : ; u 1 = : : : ; u 2 = : : : ; . . . E.5121 Determine the first five terms of the sequence u n , defined for all n ∈ N , arithmetic with first term 2 and reason 3 . E.7309 Consider the sequence u n arith-metic, defined for any n ∈ N , of first term 2 and reason − 3 . Determine the first four terms of the sequence u n . https://chingmath.fr a0×:::a1×:::a2×:::a3×:::a4×:::×:::×:::×::: b0−:::b1−:::b2−:::b3−:::b4−:::−:::−:::− chapExoCorrec/2372 sacados/2372 chapExoCorrec/7195 sacados/7195 chapExoCorrec/8523 sacados/8523 chapExoCorrec/5121 sacados/5121 chapExoCorrec/7309 sacados/7309
E.7717 1 a In a programming language, enter the following al-gorithm : a ← 2 For i ranging from 0 to 4 a ← a+3 End For b Performing a step-by-step execution, note the succes-sive values taken by the variable a : . . . ; . . . ; . . . ; . . . ; . . . ; . . . 2 a Modify the algorithm so that the successive values taken by the variable a are: 2 ; 6 ; 10 ; 14 ; 18 ; 22 b Modify the algorithm so that the successive values taken by the variable a are: 5 ; 10 ; 15 E.10234 Definition: let u n be a sequence defined for any n ∈ N and let r ∈ R . We say that the sequence u n is a arithmetic sequence of reason r if it verifies the relation: u n +1 = u n + r for any n ∈ R . Let u n be an arithmetic sequence of reason 3 . Simplify the following expressions : a u 4 + 3 b u 10 − 3 c u 7 + 6 E.7193 Consider the sequence u n , defined for n ∈ N ∗ , arithmetic with first term 3 and reason 5 . Determine the first five terms of this sequence. E.7540 Determine the first five terms of the sequence v n , defined for all n ∈ N , arithmetic with first term 3 and reason − 2 3 . E.7186 Consider the sequence v n arith-metic defined by: v 0 = 6 ; v n +1 = v n − 2 for all n ∈ N Determine the value of the first 6 terms of the sequence v n . 6. Arithmetic sequences: introduction to the explicit formula E.7191 Consider the sequence u n n ∈ N whose first terms are: u 0 =3 ; u 1 =7 ; u 2 =11 ; u 3 =15 Among the four propositions, name the two relations verified by the first four terms of the sequence u n . Which ones? a u n +1 = u n + 4 b u n +1 = 4 · u n c u n = 3 + 3 · n d u n = 3 + 4 · n E.9488 Consider the sequence u n , defined for n ∈ N , arithmetic and whose first terms have the value : n 0 1 2 3 4 u n 3 7 11 15 19 1 Give the characteristic elements of this sequence. 2 Which of the following relationships are verified by the sequence u n : a u n +1 = u n + 4 b u n +1 = u n + 3 c u n = 4 · n + 3 d u n = 3 · n + 4 7. Arithmetic sequences: explicit formulas E.10253 consider the sequence u n arith-metic with first term 5 and reason − 2 . Which of the following expressions represents the term u 27 : a u 26 + 2 b u 26 − 2 c u 26 + 5 d u 26 − 5 e − 2 + 5 × 27 f 5 − 2 × 27 E.9460 Proposition: let u n be an arithmetic sequence of first term u 0 and reason r . For any n ∈ N , we have : u n = u 0 + r · n This relationship is called the explicit formula of the terms of an arithmetic sequence. Consider the sequence u n n ∈ N defined by the recurrence re- lation: 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.10228 Consider the sequence u n , de-fined for any n ∈ N , arithmetic of first term 3 and reason 2 3 . 1 Give the explicit formula for the terms of the sequence u n as a function of n . 2 Determine the expression of the term of rank 112 of the sequence u n . https://chingmath.fr chapExoCorrec/7717 sacados/7717 chapExoCorrec/10234 sacados/10234 chapExoCorrec/7193 sacados/7193 chapExoCorrec/7540 sacados/7540 chapExoCorrec/7186 sacados/7186 chapExoCorrec/7191 sacados/7191 chapExoCorrec/9488 sacados/9488 chapExoCorrec/10253 sacados/10253 chapExoCorrec/9460 sacados/9460 chapExoCorrec/10228 sacados/10228
8. Arithmetic sequences: explicit formulas (extended) E.10236 Proposition: let u n be an arithmetic sequence of reason r and let k , n be two natural numbers. We have the rela-tion : u n = u k + n − k · r Let v n be an arithmetic sequence defined for all n ∈ N and of reason q . Complete the following expressions : a u 7 = u 3 + : : : × r b u 25 = u 11 + : : : × r c u 3 = u 8 + : : : × r d u 15 = u 23 + : : : × r E.5120 Let u n be an arithmetic sequence of reason r . Complete the following expressions : a u 12 = u 5 + : : : × r b u 57 = u 38 + : : : × r c u 3 = u 8 + : : : × r d u 23 = u 38 + : : : × r E.10229 Consider the sequence u n , de-fined for all n ∈ N ∗ , arithmetic with first term 15 and reason − 3 . 1 Give the explicit formula for the terms of the sequence u n as a function of n . 2 Determine the value of the term of rank 17 of the se-quence u n . E.9508 Consider the sequence u n , defined for any n ∈ N , arithmetic of reason 3 and whose term of rank 8 has value : u 8 =25 1 Determine the value of the term u 14 . 2 Determine value of term u 3 . E.10571 Consider the arithmetic sequence u n , defined on N , of first term 5 and reason 5 3 . Give the value of the term u 15 of rank 15 . 9. Arithmetic sequences: rank of a term E.6530 Consider the suite u n n ∈ N arith-metic with first term 3 and reason − 2 . 1 Determine the value of terms u 12 and u 43 . 2 Determine the value of rank n realizing the equalities: a u n = − 21 b u n = − 57 E.9537 Consider the sequence u n defined for all n ∈ N ∗ arithmetic with first term of 2 and reason 0.25 . Determine the rank n of the term in the sequence u n such that : u n = 15.75 E.8048 Consider the suite u n n ∈ N arith- metic with first term 4 and reason 1 3 . 1 Determine the value of the term u 8 . 2 Determine the rank n such that : u n = 16 E.8406 Consider the sequence u n defined on N arithmetic of first term 2 and reason 3 4 . Determine the rank of the term with value 53 4 E.10572 Consider the arithmetic sequence u n , defined on N ∗ , of first term − 6 and reason 3 . Determine the rank of the term of the sequence u n having value 48 . 10. Arithmetic sequences: characteristic features E.5135 Let u n n ∈ N be an arithmetic se-quence, defined for any n ∈ N , whose two terms are known : u 4 = 12 ; u 22 = − 24 Give, justifying your approach, the characteristic elements of this sequence. E.6546 Consider the sequence u n , defined for all n ∈ N ∗ , arithmetic, for which the values of the following two terms are known : u 10 = 5 ; u 16 = 14 Determine the first term u 1 and the ratio of this sequence. E.10237 Let w n n ∈ N be the arithmetic se-quence such that : w 6 = 7 ; w 8 = 1 Determine the characteristic elements of the sequence u n . E.10238 Let w n n ∈ N ∗ be the arithmetic sequence that verifies : w 15 = 54 ; w 99 = 180 Determine the characteristic elements of the sequence u n . https://chingmath.fr chapExoCorrec/10236 sacados/10236 chapExoCorrec/5120 sacados/5120 chapExoCorrec/10229 sacados/10229 chapExoCorrec/9508 sacados/9508 chapExoCorrec/10571 sacados/10571 chapExoCorrec/6530 sacados/6530 chapExoCorrec/9537 sacados/9537 chapExoCorrec/8048 sacados/8048 chapExoCorrec/8406 sacados/8406 chapExoCorrec/10572 sacados/10572 chapExoCorrec/5135 sacados/5135 chapExoCorrec/6546 sacados/6546 chapExoCorrec/10237 sacados/10237 chapExoCorrec/10238 sacados/10238
E.8524 Let v n n ∈ N ∗ be an arithmetic se-quence such that : v 7 = 13 ; v 15 = 39 Determine the value of the first term and the reason of the sequence. E.2400 Let w n n ∈ N be an arithmetic se-quence such that : w 0 = 5 ; w 9 = 25 Determine the characteristic elements of the sequence u n . E.2452 Let u n be an arithmetic sequence defined for all n ∈ N ∗ for which the following two terms are known : u 7 = 3 ; u 19 = 11 Determine the first term and reason of this sequence. E.2428 Consider the sequence u n n ∈ N arithmetic whose value of two terms is known : u 14 =2 ; u 20 =0 1 Determine the first term and reason of this sequence. 2 a Determine the expression of the term u n as a func-tion of the value of n . b Determine the rank of the term worth 10 3 E.8359 Consider a sequence u n , defined for any n ∈ N , arithmetic such that : u 2 is double u 0 ; u 6 i.e. the square of u 2 . Determine the characteristic elements of the two arithmetic sequences achieving these conditions. 11. Recognizing an arithmetic sequence E.7187 Consider the sequence u n , defined for any 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.6523 Consider the two sequences u n and v n defined for any n ∈ N and whose first terms are given be-low : u 0 =3 ; u 1 =5 ; u 2 =7 ; u 3 =10 ; u 4 =12 ; u 5 =14 v 0 =6 ; v 1 =3.5 ; v 2 =1 ; v 3 = − 1.5 ; v 4 = − 4 ; v 5 = − 6.5 For quelle (s) suite (s) , can we conjecture that the sequence is an arithmetic sequence? For quelle (s) suite (s) , can it be stated that the sequence is not arithmetic. E.9547 Consider the sequence u n defined for any integer n ∈ N by: u n = 2 + 3 × n 1 Let n ∈ N , simplify the expression u n +1 − u n . 2 Deduce the nature of the sequence u n , as well as its characteristic elements. 12. Geometric sequences: first terms E.7188 Definition: we call geometric sequence any sequence of numbers whose successor of a term is obtained by multiply-ing it by the same number. This number is called the reason of the geometric sequence. Consider the sequence u n , defined for any n ∈ N , geometric of first term 2 and reason 3 . Determine the first five terms of this sequence. E.7189 Consider the sequence v n defined by the recurrence relation: v 0 = 64 ; v n +1 = 1 2 · v n for any n ∈ N 1 What is the nature of this sequence? 2 Give, without justification, the value of v 6 . E.9496 Consider the sequence v n geomet-ric defined by: v 0 = − 2 ; v n +1 = 1 2 · v n for any n ∈ N Determine the values of the first 6 terms of the sequence v n . E.5122 The sequence v n is defined for all n ∈ N : Determine the first four terms of the suite v n geometric with first term 3 and reason − 3 2 . E.10235 Definition: let u n be a sequence defined for any n ∈ N and let q ∈ R . We say that the sequence u n is a geometric sequence of reason r if it verifies the relation: u n +1 = u n × q for all n ∈ R . Let u n be a geometric sequence of reason 3 . Simplify the following expressions : a 3 × u 10 b u 12 3 c u 5 × 3 2 E.8525 Let u n be a geometric sequence defined for n ∈ N , of reason 2 and whose first term has the value 3 8 . Determine the first six terms of this sequence. https://chingmath.fr chapExoCorrec/8524 sacados/8524 chapExoCorrec/2400 sacados/2400 chapExoCorrec/2452 sacados/2452 chapExoCorrec/2428 sacados/2428 chapExoCorrec/8359 sacados/8359 chapExoCorrec/7187 sacados/7187 chapExoCorrec/6523 sacados/6523 chapExoCorrec/9547 sacados/9547 chapExoCorrec/7188 sacados/7188 chapExoCorrec/7189 sacados/7189 chapExoCorrec/9496 sacados/9496 chapExoCorrec/5122 sacados/5122 chapExoCorrec/10235 sacados/10235 chapExoCorrec/8525 sacados/8525
E.9499 Consider the sequence v n , defined for all n ∈ N ∗ , geometric with first term 54 and reason 1 3 . De-termine the first four terms of the sequence v n . 13. Geometric sequences: explicit formula E.7272 Proposition: let u n be a geometric sequence of first term u 0 and reason q . For any n ∈ N , we have : u n = u 0 × q n This relationship is called the explicit formula of the terms of a geometric sequence. Let u n be a geometric sequence, defined for any n ∈ N with reason 2 and first term 5 Determine the value of u 8 . E.6531 Consider the sequence u n geomet-ric, defined for all n ∈ N , of first term 2 4 3 and reason 3 2 . Determine the value of terms u 11 and u 28 . E.8049 Consider the sequence u n geomet-ric, defined for any natural number n , of first term 4 and reason 2 3 . Determine, in simplified form, the value of the term u 4 . E.9538 Consider the sequence u n , defined for all n ∈ N , geometric of first term 3 4 7 6 and reason 3 × 7 2 . Give the expression for the term u 10 in the form : u 10 = 3 k × 7 where k;‘ ∈ N E.9505 Consider the sequence u n defined on N geometric of first term 2 and reason 3 4 . Determine the value of the term of rank 6 . 14. Geometric sequences: explicit formula (extended) E.10254 Consider the geometric sequence, defined on N , with first term 4 and reason 3 . Which of the following expressions represent the term u 27 : a u 5 × 3 22 b u 5 × 3 27 c u 31 × 3 4 d u 31 × 3 − 4 E.5123 Definition: let u n be an arithmetic sequence of reason r and let k , n be two natural numbers. We have the relation: u n = u k · q n − k Let v n be a geometric sequence defined for all n ∈ N and of reason q . Complete the following expressions : a u 7 = u 3 × q ... b u 25 = u 11 × q ... c u 3 = u 8 × q ... d u 15 = u 23 × q ... E.8407 Consider the suite u n defined on N ∗ geometric with first term 5 3 and reason 5 7 . Determine the expression, in simplified form, of the term of rank 7 of the sequence u n . E.9498 Let v n be a geometric sequence defined for any n ∈ N , of reason 3 2 and such that v 6 =12 . De-termine the value of v 3 . E.10575 Consider the sequence v n geo-metric of reason 2 3 3 2 and such that v 7 =108 . 1 Determine the value of v 21 . 2 Determine the value of v 5 . E.9497 Let v n be a geometric sequence, defined for n ∈ N , of reason 1 2 and such that v 7 =3 2 × 2 3 . De-termine the value of v 20 . E.7275 Let u n be a geometric sequence, defined for any n ∈ N of reason 3 and such that : u 7 = 3 2 × 2 2 Determine the value of u 2 . 15. Geometric sequences: determine the rank of a term E.10241 Proposition 1: Let m and n be two integers and a number a strictly positive such that a m = a n then m = n . Proposition 2: (admitted) Let x and y be two strictly positive numbers such that there exists a non-zero integer n such that x n = y n . If n is odd then x = y If n is even then x = y or x = − y https://chingmath.fr chapExoCorrec/9499 sacados/9499 chapExoCorrec/7272 sacados/7272 chapExoCorrec/6531 sacados/6531 chapExoCorrec/8049 sacados/8049 chapExoCorrec/9538 sacados/9538 chapExoCorrec/9505 sacados/9505 chapExoCorrec/10254 sacados/10254 chapExoCorrec/5123 sacados/5123 chapExoCorrec/8407 sacados/8407 chapExoCorrec/9498 sacados/9498 chapExoCorrec/10575 sacados/10575 chapExoCorrec/9497 sacados/9497 chapExoCorrec/7275 sacados/7275 chapExoCorrec/10241 sacados/10241
1 For each equality, determine the value of the integer n : a 2 n = 64 b 3 n = 81 c 3 4 n = 27 64 2 For each equality, determine the possible value(s) of the real number x such that : a x 3 = 1 8 b x 4 = 81 625 c x 2 = 36 49 E.9504 Consider the suite u n defined on N ∗ geometric with first term 5 3 and reason 5 7 . Determine the rank of the term with value 5 12 7 9 E.9506 Consider the sequence u n geomet-ric, defined for any natural number n , of first term 4 and reason 2 3 . Using the calculator, determine the value of rank n verifying: u n = 8 192 177 147 E.9507 Consider the suite u n geometric, defined for any n ∈ N , of first term 2 4 3 and reason 3 2 . For each question, determine the rank n realizing equality: a u n = 3 8 2 5 b u n = 3 19 2 16 E.10574 Consider the sequence u n geo-metric, defined on N ∗ , of first term 54 and reason 3 2 2 . Determine the rank of the term with value 3 27 2 11 . 16. Geometric sequences: characteristic features E.8526 Let v n be a geometric sequence, defined for any n ∈ N , of reason q and whose two terms are known : v 11 = 4 7 ; v 14 = 27 14 1 a Complete the relationship below : v 14 = v 11 × : : : b Deduce the value of the reason for the sequence v n . 2 Determine the value of the first term of the sequence v n . E.2401 Consider w n a geometric sequence defined for all n ∈ N and such that : w 0 = 5 ; w 3 = 40 Determine the reason for the sequence u n . E.10239 Consider the geometric sequence w n , defined for all n ∈ N and such that : w 3 = 3 8 ; w 6 = − 3 64 Determine the characteristic elements of the sequence w n . E.9500 Let v n , defined for any n ∈ N , be a geometric sequence whose two terms are known : v 4 = 8 ; v 7 = 64 27 Give, justifying your approach, the characteristic elements of this sequence. E.9501 Consider the sequence v n , defined for any n ∈ N ∗ , geometric whose values of the following two terms are known : v 4 = 96 ; v 7 = 3 2 Determine the first term v 1 and the reason for this sequence. E.2429 Consider the sequence u n geomet-ric defined for any n ∈ N and whose terms are known : u 5 = 2 ; u 8 = 27 4 1 Determine the first term and reason of this sequence. 2 a Give the explicit expression of the term u n as a func-tion of rank n . b Determine the rank of the term worth 16 27 E.5827 Determine the seven-term ge-ometric progressions (with real terms) such that the sum of the first three terms is equal to 2 and the sum of the last three terms is equal to 1 250 E.9502 Let v n be a geometric sequence defined for any n ∈ N and whose rank terms 4 and 8 are 3 and 16 27 respectively Determine the two possible values of the reason. Give the value of the first term of both sequences. E.2412 Let u n be a sequence defined for any n ∈ N for which we know the value of the following two terms : u 6 = 36 ; u 10 = 9 4 Show that there are at least two geometric sequences verifying these conditions. E.10240 Consider the sequence w n geo-metric defined for any n ∈ N and such that : w 124 = 2 × 10 − 4 ; w 128 = 1 8 Note that there are two sequences verifying these conditions. We will determine the characteristic elements of each of these sequences. 17. Recognizing a geometric sequence https://chingmath.fr chapExoCorrec/9504 sacados/9504 chapExoCorrec/9506 sacados/9506 chapExoCorrec/9507 sacados/9507 chapExoCorrec/10574 sacados/10574 chapExoCorrec/8526 sacados/8526 chapExoCorrec/2401 sacados/2401 chapExoCorrec/10239 sacados/10239 chapExoCorrec/9500 sacados/9500 chapExoCorrec/9501 sacados/9501 chapExoCorrec/2429 sacados/2429 chapExoCorrec/5827 sacados/5827 Bac Khmer Juin 1969 chapExoCorrec/9502 sacados/9502 chapExoCorrec/2412 sacados/2412 chapExoCorrec/10240 sacados/10240
E.9482 Consider the sequence v n , defined for any n ∈ N , whose first terms are noted in the table below : n 0 1 2 3 4 v n 96 48 24 12 6 Of the four relationships proposed, only two are verified by the first five terms of the sequence v n . Which ones? a v n +1 = 2 · v n b v n +1 = 1 2 · v n c v n = 96 × 1 2 n d v n = 1 2 × 96 n E.9503 Consider the sequence v n , defined for any n ∈ 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 an arithmetic sequence. E.6524 Consider the two sequences of num-bers below où given the first six terms : a 8 ; 4 ; 2 ; 1 ; 1 2 ; 1 4 b 1 ; 3 ; 9 ; 18 ; 54 ; 162 For each of the questions, can we conjecture that the sequence is a geometric sequence? If so, specify the first term and the reason. If not, justify your rejection of the conjecture. E.9546 Consider the sequence u n defined for any integer n ∈ N by: u n = 4 × 3 n Note: for any integer n ∈ N , u n is non-zero. This is also written as : ∀ n ∈ N , u n = 0 1 Let n ∈ N , simplify the expression u n +1 u n . 2 Deduce the nature of the sequence u n , and its charac-teristic elements. E.7274 For each question is defined a se-quence u n for any natural number n . Say whether or not this is a geometric sequence, justifying your answer and giv-ing, where appropriate, its characteristic elements : a u n = 3 · n + 1 b u n = 5 n + 5 n +1 c u n = 2 × 4 n 3 n +1 d u n = n n E.7273 Consider the two sequences u n and v n defined for any natural number n explicitly by: u n = 2 n +2 + 2 n ; v n = 4 n +1 3 n Determine the nature and characteristic elements of each of these two sequences. 18. Arithmetic sequences and modeling E.9545 Régis has decided to get back into the gym and start doing sets of ˇ abdo ı every day. He plans to increase the number of abdos to be done each day by the same number. On the day he’s due to start training, he remembers that he was supposed to do 16 abdo on 6 e day and and 40 abdos on 14 e day. Help him find the number of abs he needs to do on the first day of his recovery. Hint: any trace of research, however incomplete, will be taken into account in this question. E.9585 A car manufacturer decides to re-duce its production of combustion engine cars. Currently, its production is 80 000 cars per month, and it decides to reduce this by 3 000 cars each month. We decide to denote the cur-rent production of combustion engine cars by u 0 and denote its production after n months by u n (where n ∈ N ) . 1 Specify the nature of the sequence u n and its charac-teristic elements. 2 Give a recurrence formula and the explicit formula for the sequence u n . 3 How many months must we wait for its production of combustion engine cars to fall below 10 000 units pro-duced per month? Hint: any evidence of research, even if incomplete, will be taken into account in this question. 19. Geometric sequences and modeling E.7260 In 2012 , a country’s population is estimated at 45.5 million. This population is assumed to de-crease by 3 % each year. We note u n the population of this country at the year 2012+ n . We have thus just created a suite u n defined on N . 1 Determine the population of this country in 2013 and 2014 rounded to the nearest thousand. 2 a Give the nature and characteristic elements of this sequence. b Give the expression of the term u n as a function of its rank n . 3 Determine the population of this country in 2020 rounded to the nearest thousand. https://chingmath.fr chapExoCorrec/9482 sacados/9482 chapExoCorrec/9503 sacados/9503 chapExoCorrec/6524 sacados/6524 chapExoCorrec/9546 sacados/9546 chapExoCorrec/7274 sacados/7274 chapExoCorrec/7273 sacados/7273 chapExoCorrec/9545 sacados/9545 chapExoCorrec/9585 sacados/9585 chapExoCorrec/7260 sacados/7260
E.7024 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 model the number of films offered by a geometric sequence u n où 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 value of the rank term 6 rounded to unity. E.7270 Some geothermal power plants operate by using heat from un-derground. To harness this natural heat, several sufficiently deep wells must be dug. When building such a plant, the cost of drilling the first well is modeled as follows u n defined for any natural integer n not equal to zero, by: u n = 2000 × 1 ; 008 n − 1 where u n represents the cost in euros of drilling the n th ten meters. We thus have u 1 =2000 and u 2 =2016 , i.e., drilling the first ten meters costs 2 000 euros, and drilling the next ten meters costs 2 016 euros. Throughout the exercise, round the results to two decimal places. 1 Calculate u 3 and then the total cost of drilling the first 30 meters. 2 For any natural number n not equal to zero: a Express u n +1 in terms of u n and specify the nature of the sequence u n . b Deduce the percentage increase in the cost of drilling the ( n +1) th ten meters compared to that of the n th ten meters. E.2928 Below are the first six ˇ flakes of Helge Von Koch ı representing one of the simplest fractals : To move from one construction to the next, we perform the following manipulation on each segment : Each segment is divided into three equal parts (step 1) . An equilateral triangle is constructed on the middle segment (step 2) . We delete the middle segment (step 3) . 1 a Moving from step n o 0 to step n o 1 reveals an equilat-eral triangle. Highlight this triangle in red. b How many segments does the figure in step n o 1 in-clude? How many equilateral triangles will appear in step n o 2 ? Highlight these triangles in red. 2 Note u n the numerical sequence whose rank term n is the number of segments making up the figure at step n ième : a Justify with a sentence that the sequence u n verifies the relation: u n +1 = 4 · u n b Express the term u n in terms of its rank n . c How many segments does the figure in step n o 5 com-prise? 3 It is assumed that the initial [ AB ] segment has length 1 . Note v n the numerical sequence whose term of rank n is the length of the polygon line forming the figure at step n ième : a Justify with a sentence that the sequence v n verifies the relation: v n +1 = 4 3 · v n b Express the term v n in terms of its rank n . 20. Recognizing arithmetic and geometric sequences E.5859 The three sequences below are de-fined for any natural integer n : 1 Briefly justify that the first terms of the sequence u n presented below can be the terms of an arithmetic se-quence whose reason is specified : u 0 = 2 ; u 1 = 9 2 ; u 2 = 7 ; u 3 = 19 2 2 Briefly justify that the first terms of the sequence v n presented below can be the terms of a geometric sequence whose reason is specified : v 0 = 24 ; v 1 = 6 ; v 2 = 3 2 ; v 3 = 3 8 3 Briefly justify that the first terms of the sequence w n represent neither the first terms of an arithmetic se-quence, nor the first terms of a geometric sequence w 0 = 1 ; w 1 = 2 ; w 2 = 4 ; w 3 = 16 https://chingmath.fr chapExoCorrec/7024 sacados/7024 chapExoCorrec/7270 sacados/7270 chapExoCorrec/2928 sacados/2928 ABEtape no0 ABEtape no1 ABEtape no2 ABEtape no3 ABEtape no4 ABEtape no5 Etape0Etape1Etape2Etape3 chapExoCorrec/5859 sacados/5859
E.2402 Below are the first four terms of five sequences defined for any natural number: 1 u 0 = 3 ; u 1 = 7 ; u 2 = 11 ; u 3 = 15 2 v 0 = 54 ; v 1 = 6 ; v 2 = 2 3 ; v 3 = 2 27 3 w 0 = 2 ; w 1 = − 6 ; w 2 = 18 ; w 3 = − 54 4 a 0 = 3.25 ; a 1 = 5 ; a 2 = 6.75 ; a 3 = 8.25 5 b 0 = 2 ; b 1 = 4 ; b 2 = 8 ; b 3 = 16 For which of these sequences can we conjecture that it is arith-metic? geometric? or can we assert that it is neither arith-metic nor geometric? E.1098 Consider the two sequences u n and v n , where their ranks are positive or zero natural numbers ( n ∈ N ) , defined by the following relations : u n = n 2 − 2 · n + 2 v 0 = 5 ; v n +1 = v n 2 − 3 · v n 1 Determine the first four terms of each of these two se-quences. 2 Justify that these two sequences are neither arithmetic nor geometric sequences. E.7307 1 Consider the sequence u n defined by: u n = n 2 + n + 2 for any integer n ∈ N Establish that the sequence u n is not a geometric se-quence. 2 Consider the sequence v n defined by: v n = 1 n 2 + 2 for any integer n ∈ N Establish that the sequence v n is not an arithmetic sequence. 21. Unclassified financial years E.10343 The two parts are independent : Part A Fabrice owns a nursery and pots a sapling measuring 54 cm, 8 months after the tree has made its first leaf. He gave it to his son Alex 4 months later, when the tree mea-sured 76 cm. Alex’s father explains that, every month, the tree’s height increases steadily from the moment it makes its first leaf until it reaches its maximum height. We note h n the height, in centimeters, of the tree n months after its first leaf. Thus, h 0 is the height of the tree when it makes its first leaf. 1 What are the two terms of the sequence ( h n ) given in the statement? (specify their value and rank) 2 What is the nature of the sequence ( h n ) ? Find its char-acteristic elements. 3 Deduce the expression of ( h n ) as a function of n . 4 a Calculate h 24 and interpret the result. b Copy and complete the Python program below, which should display the value of the term h 24 in console : h=... for i in range(...): h = ... print(h) 5 Alex will have to replant the tree in the ground as soon as its height exceeds 1.60 m . Determine the number of months required after its first leaf appears. Part B Let be the sequence defined on N by u n = 3 n 2 + 16 n + 5 n + 5 1 Factor the numerator. 2 Simplify the expression of the term u n . 3 Prove that the sequence is arithmetic. Give its first term and its reason. E.10566 Consider the step-by-step pattern whose first 4 patterns are shown below : 1 How many stitches does it take to build the pattern n o 4? the pattern n o 5? 2 We note u n the number of points required to construct the number of points in the pattern n . a Give the recurrence formula defining the sequence u n . b Give the explicit formula defining the sequence u n . E.10567 Consider the step-by-step pattern construction whose first five steps are shown below : 1 How many tiles are needed to construct pattern 4? pat-tern 5? 2 We note u n the number of tiles required to construct pattern n . a Give the recurrence formula verified by the sequence u n . b Give the explicit formula verified by the sequence u n . Hint: we’ll admit that the explicit formula is a polynomial of the second degree. https://chingmath.fr chapExoCorrec/2402 sacados/2402 chapExoCorrec/1098 sacados/1098 chapExoCorrec/7307 sacados/7307 chapExoCorrec/10343 sacados/10343 From Maud Saveur sacados/10566 Motif0Motif1Motif2Motif3 chapExoCorrec/10567 sacados/10567 Motif0Motif1Motif2Motif3
E.10568 Consider the step-by-step pattern construction whose first five steps are shown below : 1 How many tiles are needed to construct pattern 4? pat-tern 5? 2 We note u n the number of tiles required to construct pattern n . a Give the recurrence formula verified by the sequence u n . b Give the explicit formula verified by the sequence u n . Hint: we’ll admit that the explicit formula is a polynomial of the third degree. https://chingmath.fr sacados/10568 Motif0Motif1Motif2Motif3