Grade 11 / Arithmetic, geometric and other sequences 94 exercises (100% 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 chapExoCorrec/2905 sacados/2905 chapExoCorrec/2379 sacados/2379 chapExoCorrec/6517 sacados/6517 chapExoCorrec/7308 sacados/7308 chapExoCorrec/6516 sacados/6516
1234567ABCDTempsPopulationdelasoucheAPopulationdelasoucheBPopulationtotale0200300 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 cul-ture 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
a0×:::a1×:::a2×:::a3×:::a4×:::×:::×:::×::: b0−:::b1−:::b2−:::b3−:::b4−:::−:::−:::− 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 . 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 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 . 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 . https://chingmath.fr chapExoCorrec/10236 sacados/10236 chapExoCorrec/5120 sacados/5120 chapExoCorrec/10229 sacados/10229 chapExoCorrec/9508 sacados/9508 chapExoCorrec/6530 sacados/6530 chapExoCorrec/9537 sacados/9537 chapExoCorrec/8048 sacados/8048 chapExoCorrec/8406 sacados/8406 chapExoCorrec/5135 sacados/5135 chapExoCorrec/6546 sacados/6546 chapExoCorrec/10237 sacados/10237 chapExoCorrec/10238 sacados/10238 chapExoCorrec/8524 sacados/8524 chapExoCorrec/2400 sacados/2400
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. 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 https://chingmath.fr 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 chapExoCorrec/9499 sacados/9499
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.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 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 https://chingmath.fr 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/9497 sacados/9497 chapExoCorrec/7275 sacados/7275 chapExoCorrec/10241 sacados/10241 chapExoCorrec/9504 sacados/9504 chapExoCorrec/9506 sacados/9506
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 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 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. https://chingmath.fr chapExoCorrec/9507 sacados/9507 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 chapExoCorrec/9482 sacados/9482 chapExoCorrec/9503 sacados/9503
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.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. 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. https://chingmath.fr chapExoCorrec/6524 sacados/6524 chapExoCorrec/9546 sacados/9546 chapExoCorrec/7274 sacados/7274 chapExoCorrec/7273 sacados/7273 chapExoCorrec/9585 sacados/9585 chapExoCorrec/7260 sacados/7260 chapExoCorrec/7024 sacados/7024
ABEtape no0 ABEtape no1 ABEtape no2 ABEtape no3 ABEtape no4 ABEtape no5 Etape0Etape1Etape2Etape3 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 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? https://chingmath.fr 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 chapExoCorrec/2402 sacados/2402
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. https://chingmath.fr chapExoCorrec/1098 sacados/1098 chapExoCorrec/7307 sacados/7307 chapExoCorrec/10343 sacados/10343 From Maud Saveur