Outside the high school program / Sequences: reasoning by recurrence 55 exercises (100% corrected)

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1. Algebraic manipulations (for heredity) E.5810 Establish the following equality for any natural number n different from 6 : 1 3 · n + 4 − 6 n − 6 = 1 3 E.5805 Establish the identity below for any nat-ural number n : 1 2 − n n + 1 = n + 1 ( n + 1) + 1 E.5811 Establish the following identity for any natural number n : 10 · n + 1 n + 10 − 1 10 · n + 1 n + 10 + 1 = 9 11 × n − 1 n + 1 E.5809 Establish the following identity for any natural number n : n · 1 + (0.5) n n + 1 2 · ( n + 1) = 1 + (0.5) n +1 n + 1 E.5807 Establish the following identity for any natural number n : ( n + 2) · 1 + 2 · n + 1 n + 1 = 1 + 2 · ( n + 1) E.5808 Establish the following identity for any natural number n : n · n + 1 · 2 n + 1 6 + ( n + 1) 2 = ( n + 1)( n + 2)(2 n + 3) 6 E.5806 Establish the following identity for any natural number n : 1 + 2 n + 1 · 4 · n 2 + 12 · n + 5 + 6 n + 1 = 4 · ( n + 1) 2 + 12 · ( n + 1) + 5 E.6168 Establish the following equality for any natural number n : 3 × 3 n 3 n + 1 1 + 2 × 3 n 3 n + 1 = 3 n +1 3 n +1 + 1 2. Introduction to reasoning by recurrence E.3437 1 Let u n be a sequence whose rank term n is defined, for any natural number n , by: u n = 2 · n n + 1 Give the simplified expression of the terms u n +1 and u n +2 as a function of n . 2 Let v n be a sequence whose rank term n can be written as a function of n : v n = 3 n − 1 + 4 n +1 for any natural number n . Give an expression for v n +1 and v n +2 as a function of n . 3 For any non-zero natural number n , we have the equal-ity: 1 2 + 2 2 + · · · + n 2 = n · ( n + 1) · (2 n + 1) 6 Give the writing of this identity at rank ( n +1) . E.6129 Consider the sequence defined by: u 0 = 7 ; u n +1 = 1 4 · u n + 3 2 1 Calculate the values of the first six terms of the se-quence u n , then their values rounded to the nearest thousandth. 2 Note that these first terms verify the property: u n 2 for all n ∈ N . How can we justify that all the terms of the sequence u n verify this property? E.5176 Consider the sequence u n n ∈ N defined by the recurrence relation and verifying the conditions : u 0 = 1 ; u 1 = 4 ; u n +2 = 2 · u n +1 − u n pour all n ∈ N 1 a Déterminer the first five terms of the sequence u n . b Make a conjecture as to the nature of the sequence u n . c What remains to be shown to establish this recurrence? 2 a Give the simplified expression of u n +2 − u n +1 . b Is this enough to justify the conjecture? E.3430 Let the sequence u n be defined for any natural number n by: u 0 = 1 2 ; u n +1 = 1 2 · u n + 2 u n for any n ∈ N 1 Consider the sequence f defined on 0 ; + ∞ by: f ( x ) = 1 2 · x + 2 x Draw up the table of variation of the function f . 2 Can we conjecture a minoration of the sequence u n for terms of rank greater than or equal to 2 . 3. Recurrence - inequalities https://chingmath.fr chapExoCorrec/5810 sacados/5810 chapExoCorrec/5805 sacados/5805 chapExoCorrec/5811 sacados/5811 chapExoCorrec/5809 sacados/5809 chapExoCorrec/5807 sacados/5807 chapExoCorrec/5808 sacados/5808 chapExoCorrec/5806 sacados/5806 chapExoCorrec/6168 sacados/6168 chapExoCorrec/3437 sacados/3437 chapExoCorrec/6129 sacados/6129 chapExoCorrec/5176 sacados/5176 chapExoCorrec/3430 sacados/3430
E.5802 Consider the numerical sequence u n defined on N by: u 0 = 1 ; u n +1 = 9 6 − u n for all n ∈ N Demonstrate by recurrence that, for any natural number n , we have the framing : 0 <u n < 3 . E.3428 Consider the sequence u n n ∈ N defined by: u 0 = 3 ; u n +1 = 3 2 · u n − 1 for all n ∈ N 1 Establish that, for any natural number n , we have : u n 3 2 Deduce that the sequence u n is increasing. E.5033 Consider the sequence u n defined on N by: u 0 = − 3 ; u n +1 = 3 + u n 2 for all n ∈ N . 1 Determine the value of the term of rank 1 of the sequence u n . 2 Show that, for any n ∈ N ∗ , we have the frame : 0 u n 2 E.3286 The sequence u n is defined by: u 0 = 1 ; u n +1 = 1 2 u n + n − 1 , for all n ∈ N . Demonstrate that for any n 3 , we have : u n 0 . E.4102 1 Show that for any natural number greater than or equal to 3 , we have : 2 · n 2 n +1 2 2 Show by recurrence that for any integer n greater than or equal to 4 : 2 n n 2 4. Recurrence - equalities E.6130 Consider the sequence u n de-fined on N by: u 0 = 0 ; u n +1 = u n + 2 n + 2 for all n ∈ N Show, using recursive reasoning, that the term of rank n of the sequence u n can be expressed as : u n = n 2 + n E.3295 Consider the sequence u n of natural numbers defined by: u 0 = 14 u n +1 = 5 u n − 6 for any n ∈ N Show by recurrence that, for any natural number n : 2 u n = 5 n +2 + 3 . E.6131 Consider the sequence u n de-fined for any natural number n by: u 0 = 2 ; u n +1 = 1 5 · u n + 3 × 0.5 n . Establish, using reasoning by recurrence, the following equal-ity for any natural number n : u n = − 8 × 1 5 n + 10 × 0.5 n E.6827 Let u n be the sequence defined by its first term u 0 =5 and, for any natural number n by: u n +1 = 0.5 · u n + 0.5 · n − 1.5 Using reasoning by recurrence, show that : u n = 10 × 0.5 n + n − 5 E.6950 Consider the sequence u n defined by: u 0 = 3 ; u n +1 = 9 × 2 n − u n for all n ∈ N Show, using recursive reasoning, that the sequence u n is a geometric sequence, specifying its first term and common ratio. E.3292 Consider the sequence u defined by: u 0 = 0 u n +1 = 1 2 − u n for all n ∈ N . . Demonstrate, using reasoning by recurrence, that for any nat-ural number n , we have : un = n n + 1 . E.5801 Consider the sequence u n de-fined by: u 1 = 3 2 ; u n +1 = n · u n + 1 2( n + 1) for all integers n ∈ N ∗ Show, using recursive reasoning, that for all non-zero natural integers n , we have : u n = 1 + (0.5) n n E.3438 For x =1 , show that for all n ∈ N , we have : 1 + x + x 2 + · · · + x n = 1 − x n +1 1 − x E.3425 Establish the following property, using reasoning by recurrence for any n 1 : 1 2 + 2 2 + 3 2 + · · · + n 2 = n ( n + 1)(2 n + 1) 6 E.6152 Consider the sequence w n whose terms verify, for any natural number n 1 : w 0 = 1 ; n · w n = n + 1 · w n − 1 + 1 for any n ∈ N ∗ Demonstrate by recurrence that we have the following rela-tionship for any non-zero natural number n : w n +1 − w n = 2 https://chingmath.fr chapExoCorrec/5802 sacados/5802 Extrait du bac Liban Mai 2013 chapExoCorrec/3428 sacados/3428 chapExoCorrec/5033 sacados/5033 chapExoCorrec/3286 sacados/3286 Extrait d'Antilles-Guyane - Septembre 2005 chapExoCorrec/4102 sacados/4102 chapExoCorrec/6130 sacados/6130 chapExoCorrec/3295 sacados/3295 chapExoCorrec/6131 sacados/6131 chapExoCorrec/6827 sacados/6827 Extrait d'Antilles-Guyane Juin 2015 chapExoCorrec/6950 sacados/6950 chapExoCorrec/3292 sacados/3292 chapExoCorrec/5801 sacados/5801 chapExoCorrec/3438 sacados/3438 chapExoCorrec/3425 sacados/3425 chapExoCorrec/6152 sacados/6152
1erétape2ièmeétape3ièmeétape E.4307 Consider the sequence u n de-fined, for any n ∈ N , by: u 0 = 0 ; u 1 = 1 ; u n 2 = u n +1 + u n 2 for n 0 . Demonstrate by using reasoning by recurrence that, for any natural number n : u n +1 = − 1 2 · u n + 1 E.4645 Consider the sequence u n defined on N by the following recurrence relation: u 0 = 0 ; u n +1 = u n − 2 n + 11 for all n ∈ N 1 Using the software of your choice, plot the scatterplot associated with the first 15 terms of this sequence. 2 Make a conjecture as to the nature of the curve passing through these points. 3 Using three selected points on this curve, determine the expression of the function f realizing the equality below for the three abscissas of these points : u n = f ( n ) 4 Establish, by recurrence, the expression of the terms of the sequence u n as a function of their rank n . E.6133 Consider the following constructions : Note u n the numerical sequence defined on N ∗ where u n represents the number of matches required to construct the n ième step. 1 Determine a recurrence relationship between a term in the sequence u n and its predecessor. 2 Demonstrate, using reasoning by recurrence, that the term of rank n of the sequence u n admits as expres-sion : u n = 2 · n 2 + 2 · n 5. Recurrence - problems E.3290 Consider the function f defined by: f ( x ) = 2 x + 1 x + 1 We admit the following properties of the function f : The function f is increasing on 1 ; 2 If x ∈ 1 ; 2 , then we have f ( x ) ∈ 1 ; 2 We define the sequence ( v n ) defined by: v 0 =2 ; v n +1 = f v n for all n ∈ N . Establish, by reasoning by recurrence, the following two prop-erties of the suite v n : 1 For any natural number n : 1 v n 2 . 2 For any natural number n : v n +1 v n . E.5733 Consider the sequence u n de-fined by: u 0 = 2 ; u n +1 = 1 + 3 u n 3 + u n for any natural number n . All terms in this sequence are assumed to be definite and strictly positive. 1 Demonstrate by recurrence that, for any natural number, we have : u n > 1 . 2 a Establish that, for any natural number n , we have : u n +1 − u n = (1 − u n )(1 + u n ) 3 + u n b Determine the direction of variation of the sequence u n . E.3279 We place ourselves in an orthonor-mal reference frame and, for any natural number n , we define the points A n by their coordinates ( x n ; y n ) as follows : x 0 = − 3 y 0 = 4 ; x n +1 = 0.8 · x n − 0 ; 6 · y n y n +1 = 0.6 · x n + 0.8 · y n For any natural number n , show that the point A n belongs to the circle with center O and radius 5 . E.5736 Let the numerical sequence u n be defined on N by: u 0 = 2 ; u n +1 = 2 3 · u n + 1 3 · n + 1 for all n ∈ N 1 a Calculate u 1 , u 2 , u 3 and u 4 , then give their values rounded to the nearest 10 − 2 . b Formulate a conjecture about the direction of variation of this sequence. 2 a Show that for any natural number n : u n n + 3 b Show that for any natural number n : u n +1 − u n = 1 3 · n + 3 − u n c Deduce a validation of the previous conjecture. https://chingmath.fr chapExoCorrec/4307 sacados/4307 chapExoCorrec/4645 sacados/4645 chapExoCorrec/6133 sacados/6133 1erétape2ièmeétape3ièmeétape chapExoCorrec/3290 sacados/3290 chapExoCorrec/5733 sacados/5733 Extrait d'Asie Juin 2013 chapExoCorrec/3279 sacados/3279 chapExoCorrec/5736 sacados/5736
E.3423 Consider the sequence w n whose terms satisfy, for any integer n 1 : w 0 = 1 ; n · w n = n + 1 · w n − 1 + 1 for all n ∈ N ∗ . 1 Complete the table of values for the sequence w n be-low : w 0 w 1 w 2 w 3 w 4 w 5 w 6 1 2 a Make a conjecture about the nature of the sequence w n and its characteristics. b Write the recurrence relation giving n · w n for rank ( n +1) . c Establish, using recursive reasoning, that the sequence w n is arithmetic; specify the characteristic elements of this sequence. E.3419 Consider the sequence u n n ∈ N defined by: u 0 = 5 u n = 1 + 2 n · u n − 1 + 6 n for any integer n 1 1 a Calculate u 1 . b The values of u 2 , u 3 , u 4 , u 5 , u 6 , u 7 , u 8 , u 9 , u 10 , u 11 are respectively equal to : 45 , 77 , 117 , 165 , 221 , 285 , 357 , 437 , 525 , 621 . From this data conjecture the nature of the sequence d n n ∈ N defined by: d n = u n +1 − u n . 2 Consider the arithmetic sequence v n n ∈ N of reason 8 and first term v 0 =16 . Justify that the sum of the n first terms of this sequence is equal to 4 n 2 +12 n . 3 Demonstrate by recurrence that for any natural number n , we have : u n = 4 n 2 + 12 n + 5 4 Validate the conjecture made in question 1 b . E.6041 Consider the sequence u n defined for any non-zero natural number n defined by: u 1 = 7 ; u n +1 = n + 2 n · u n − 3 for all n ∈ N ∗ 1 Let v n be the sequence defined by: v n = u n n for n ∈ N ∗ . Show that the sequence v n is an arithmetic sequence whose first term and reason are to be specified. 2 Deduce an expression for the terms of the sequence u n as a function of n . 6. Strong recurrence E.6203 Consider the sequence defined by the relations : u 0 =3 ; u 1 =8 ; u n +1 =5 · u n − 6 · u n − 1 for any n ∈ N ∗ 1 Determine the values of the terms u 2 and u 3 . 2 Using reasoning by recurrence, show that for any non-zero natural number n , we have : u n = 2 n + 2 × 3 n E.6867 Consider the sequence u n defined for any natural number n by: u 0 = 0 ; u 1 = 2 ; u n +1 = 4 · u n − 4 · u n − 1 ∀ n ∈ N ∗ Show, using reasoning by recurrence, that the terms of the sequence u n admit as expression : u n = n · 2 n E.3439 Using reasoning by recurrence, show the following equality: n k =1 ( − 1) k · k 2 = ( − 1) n · n k =1 k 7. Limits and recurrences E.5819 Part A Consider the function f defined on R + by: f ( x ) = x + 2 2 · x + 1 1 Determine the limit of the function f in + ∞ . 2 Draw up the table of variations of the function f . 3 Justify that the function f is strictly positive on R + Part B Consider the sequence u n defined on N by: u 0 = 2 ; u n +1 = u n + 2 2 · u n + 1 for any integer n ∈ N 1 Using reasoning by recurrence, establish that for any nat- ural number n , we have the frame : 1 2 u n 2 2 a Establish that for any natural number n : u n +1 − 1 = − u n + 1 2 · u n + 1 b Demonstrate by recurrence that for any natural num-ber n , u n − 1 has the same sign as ( − 1) n . https://chingmath.fr chapExoCorrec/3423 sacados/3423 chapExoCorrec/3419 sacados/3419 chapExoCorrec/6041 sacados/6041 chapExoCorrec/6203 sacados/6203 chapExoCorrec/6867 sacados/6867 chapExoCorrec/3439 sacados/3439 chapExoCorrec/5819 sacados/5819
234567I2JOCf E.5748 Consider the function f defined on R + by the following relationship : f ( x ) = 3 · x 1 + 2 · x Note C f the representative curve of the function f in a refer-ence frame O ; I ; J whose representation is given below : 1 a Justify that the function f is increasing on R + . b Justify that the curve C f admits an asymptote at + ∞ . c Determine the reduced equation of the tangent ( T ) to the curve C f at the point of abscissa 1 . d Consider the line ( d ) , first bisector of the plane, of equation y = x . Study the position of the curve C f relative to the line ( d ) . 2 Consider the sequence u n defined by the relations : u 0 = 1 2 ; u n +1 = 3 · u n 1 + 2 · u n a Demonstrate, using reasoning by recurrence, that we have the following framing : 0 u n u n +1 1 for any natural number n . b Demonstrate, using reasoning by recurrence, the fol-lowing equality: u n = 3 n 3 n + 1 for any natural number n . c Deduce the limit of the sequence u n . E.3583 We define the real sequences u n n 0 and v n n 0 by: u 0 = 2 ; for all n ∈ N u n +1 = u n 2 + 5 2 · u n v n = u n − 5 u n + 5 We assume that the sequences u n and v n are defined on N . That is, for every natural number n , we have : u n =0 ; v n = − 5 1 Show that for all n 0 , we have v n +1 = v n 2 . Deduce the relation: v n = v 0 (2 n ) for all n 0 . 2 Show that v 0 = − 1 2+ 5 2 and deduce the upper bound : | v 0 | 1 16 . Then determine the limit of the sequence v n n 0 , then that of the sequence u n n 0 when n tends towards + ∞ . E.3420 We define : the sequence u n by: u 0 = 13 u n +1 = 1 5 · u n + 4 5 for any natural number n the sequence S n , for any natural number n , by: S n = n k =0 u k = u 0 + u 1 + · · · + u n 1 Show by recurrence that, for any natural number n : u n = 1 + 12 5 n . Deduce the limit of the sequence u n . 2 a Determine the direction of variation of the sequence S n . b Calculate S n as a function of n . c Determine the limit of the sequence S n . E.5831 Consider the sequence u n n ∈ N defined by: u 0 = 1 ; u n +1 = 1 3 · u n + n − 2 for everything n ∈ N 1 Using reasoning by recurrence, establish that, for any natural number n , we have : u n = 25 4 · 1 3 n + 3 2 · n − 21 4 2 Let S n be the sum defined for any natural number n by S n = n k =0 u k . Determine the expression of S n as a function of n . 3 Consider the sequence L n defined by: L n = S n n for any natural number n . Determine the value of the limit of the sequence L n . E.3708 1 Let the numerical sequences ( u n ) and ( v n ) be defined for any n ∈ N by: u 0 = 1 ; for any n ∈ N u n +1 = u n − 2 v n = 2 u n a Show that the sequence ( v n ) is geometric. Determine its characteristic elements. b We pose for all n ∈ N ∗ : S n = v 0 + v 1 + ··· + v n Set the following limit: lim n ↦→ + ∞ S n = 8 3 2 We define the sequence x n defined by: x 0 = 0 ; x n +1 = 1 2 x n + 3 2 for all n ∈ N a Show by recurrence that for any non-zero natural num-ber n : x n = 3 2 + 3 2 2 + · · · + 3 2 n b Deduce a simple expression for x n as a function of n . c Determine the limit of the sequence ( x n ) . https://chingmath.fr chapExoCorrec/5748 sacados/5748 234567I2JOCf chapExoCorrec/3583 sacados/3583 chapExoCorrec/3420 sacados/3420 Extrait de Antilles-Guyanes Septembre 2008 chapExoCorrec/5831 sacados/5831 chapExoCorrec/3708 sacados/3708
......G2...P2G1......G2...P2P1 pn...Gn...PnGn......Gn...PnPn AnAnAnAnAnAn E.3424 Let u n n ∈ N , be the sequence defined by the following recurrence relation: u 0 = 0 ; u 1 = 1 u n +2 = 4 3 · u n +1 − 1 3 · u n pour tout n ∈ N 1 Determine the exact value of the first five terms of the sequence u n . 2 We define the sequence v n n ∈ N ∗ defined by: v n +1 = u n +1 − u n for any natural number n . a Determine the first four terms of the sequence v n . b Show that the sequence v n is a geometric sequence ; its characteristic elements will be specified. 3 By reasoning by recurrence, show that for any n ∈ N , we have : u n = 3 2 · 1 − 1 3 n 4 Deduce the limit: lim n ↦→ + ∞ u n . 8. Sequences and probabilities E.6202 Pierre and Claude are playing ten-nis. Both players have the same chance of winning the first game. Subsequently, when Pierre wins a game, the probabil-ity that he will win the next one is 0.7 . And if he loses a game, the probability that he will lose the next one is 0.8 . Throughout the exercise, n is a non-zero natural number. We consider the events : G n : ˇ Pierre wins the n th game ı. P n : ˇ Pierre loses the n th game ı. For any non-zero natural number n , we set : p n = P ( G n ) . 1 Let’s study the first two games : a Complete the probability tree below : b Determine the probability of event G 2 . c Given that Pierre won the second game, what is the probability that Claude won the first game? 2 Let’s continue to study the games between Pierre and Claude : a Complete the probability tree below : b Deduce the relationship : p n +1 = 0.5 × p n + 0.2 for all n ∈ N ∗ We consider the sequence v n defined for all non-zero natural numbers n by the relationship : v n = p n − 2 5 . c Prove that the sequence v n is a geometric sequence with common ratio 0.5 d Deduce an expression for the terms of the sequence v n , then for the terms of the sequence p n in terms of n . e Determine the limit of p n when n tends to + ∞ . f How can we interpret the value of the limit of the se-quence p n in relation to the statement of the exer-cise? E.6813 In a probabilized space Ω ; P . Con-sider a sequence of events A n verifying the following rela-tions : P A 0 ) = 0.4 ; P A n A n +1 = 0 ; 6 P A n A n +1 = 0.4 for any n ∈ N We note : p n = P A n . 1 Complete the probability tree opposite. 2 Establish that : p n +1 = 0.2 · p n + 0.4 3 Demonstrate by recurrence that the terms of the se-quence p n admit for any natural number n : p n = − 0.1 × 0.2 n + 0.5 9. A little further: reasoning by recurrence E.3287 Consider the sequence u n de-fined by: u n = n 10 2 n for any natural number n . We admit that for any natural number n greater than or equal to 16, we have : 1 + 1 n 10 1.9 https://chingmath.fr chapExoCorrec/3424 sacados/3424 chapExoCorrec/6202 sacados/6202 ......G2...P2G1......G2...P2P1 pn...Gn...PnGn......Gn...PnPn chapExoCorrec/6813 sacados/6813 AnAnAnAnAnAn chapExoCorrec/3287 sacados/3287
Show, using reasoning by recurrence that for any natural num-ber n greater than or equal to 16 , we have the following frame : 0 u n 0.95 n − 16 · u 16 E.6206 Using reasoning by recurrence, establish that the equality below holds for any non-zero natural num-ber n : 1 3 + 3 3 + · · · + (2 n − 1) 3 = 2 · n 4 − n 2 E.3440 Show, using reasoning by recur-rence, that for any non-zero natural number n and for any number x belonging to the interval − 1 ; + ∞ , we have the following property: 1 + x n 1 + n · x E.3478 Prove by recurrence that for any non- zero natural number n , we have : n k =1 k 3 = n k =1 k 2 E.6037 Consider the two sequences u n and v n defined for any integer n ∈ N ∗ by: u n = n k =1 k ; v n = n k =1 k 2 1 Study the suite w n defined for any n ∈ N ∗ by: w n = v n u n 2 Deduce an expression for the sequence v n as a function of n . 10. Unclassified financial years E.6952 For each question, a statement is pro-vided. Indicate whether it is true or false and justify your answer. Any answer without justification will not be consid-ered. Question 1 Consider the sequence u n defined by: u 0 = 1 ; u n +1 = 2 · u n + 2 · n + 1 Statement : The sequence u n is a geometric sequence. Question 2 Consider the sequence u n defined by: u 0 = 0 ; u n +1 = u n + 2 · n + 2 Statement : For any natural number n , we have : u n = n 2 + n Question 3 Consider the geometric sequence u n with first term 4 and common ratio 1 2 . Consider the following algorithm: u ← 4 n ← 0 S ← u As long as 8 − S 10 − 2 u ← 0.5 × u n ← n+1 S ← S+u End while Assertion : At the end of the algorithm execution, the variable n has the value 10 . https://chingmath.fr chapExoCorrec/6206 sacados/6206 chapExoCorrec/3440 sacados/3440 chapExoCorrec/3478 sacados/3478 chapExoCorrec/6037 sacados/6037 chapExoCorrec/6952 sacados/6952