Grade 12 / Limits of sequences and reasoning by recurrence 34 exercises (including 32 corrected)

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1. Recursive reasoning E.3642 Organized retrieval of knowl-edge. We will assume that the following results are known : Proposition: e 0 = 1 . For all real numbers x and y : e x × e y = e x + y . 1 Prove that for all real numbers x : e − x = 1 e x . 2 Prove that for all real numbers x and for all natural num-bers n : e x n = e nx E.3494 Using reasoning by recurrence, show that for any non-zero natural number n , we have : x n − 1 = x − 1 1 + x + x 2 + · · · + x n − 1 2. Explicit formula for a sequence E.4223 The sequence u n is defined by u 0 =13 and for any natural number n , we have : u n +1 = 1 5 · u n + 4 5 1 Show by recurrence that, for any natural number: u n = 1 + 12 5 n 2 Deduce the limit of the sequence u n . E.3660 Consider the sequence ( u n ) n ∈ N defined by: u 0 = 5 ; u n = 1 + 2 n u n − 1 + 6 n for all n ∈ N ∗ 1 Calculate u 1 , u 2 , u 3 . 2 Demonstrate by recurrence that for any natural number n , we have : u n =4 n 2 +12 n +5 3 Consider the sequence ( d n ) n ∈ R defined by: d n = u n +1 − u n Determine the nature of the sequence (specify its first term and reason) . 3. Explicit formula, comparison and variations E.6713 Let u n be the sequence defined by u 0 =2 and, for any natural number n , by: u n +1 = 2 · u n + 2 · n 2 − n 1 Demonstrate, using reasoning by recurrence, that the terms of the sequence u n are strictly positive. 2 Demonstrate that the sequence u n is strictly increasing on N . Consider also the sequence v n defined, for any natural num-ber n , by: v n = u n + 2 · n 2 + 3 · n + 5 3 Establish that the sequence v n is a geometric sequence of reason 2 . 4 Deduce an expression for the terms of the sequence u n . E.3640 Let u n be a sequence defined by: u 0 = 1 ; u n +1 = 1 3 u n + n − 2 for all n ∈ N 1 Using reasoning by recurrence, show that for any natural number n 4 : u n 0 . 2 We define the sequence v n n ∈ N by: v n = − 2 u n + 3 n − 21 2 for all n ∈ N a Show that the sequence v n n ∈ N is a geometric se-quence whose reason and first term will be given. b Deduce that for any n ∈ N , we have : u n = 25 4 · 1 3 n + 3 2 · n − 21 4 4. Sum of the terms of a sequence E.6097 Consider the sequence u n de-fined by: u 0 = 0 ; u n +1 = u n + 2 · n + 2 for any n ∈ N . We define, for any natural number n , the sequence v n : v n = u n +1 − u n 1 a Express v n as a function of the natural number n . b Justify that the sequence v n is an arithmetic se- quence of reason 2 and whose first term will be speci-fied. 2 We define the sequence S n by: S n = n k =0 v k = v 0 + v 1 + · · · + v n for any n ∈ N . Demonstrate, using reasoning by recurrence, that : S n = ( n + 1)( n + 2) for any n ∈ N . https://chingmath.fr chapExoCorrec/3642 sacados/3642 Liban Juin 2010 chapExoCorrec/3494 sacados/3494 chapExoCorrec/4223 sacados/4223 Extrait d'Antilles-Guyane Septembre 2008 chapExoCorrec/3660 sacados/3660 La reunion Juin 2008 chapExoCorrec/6713 sacados/6713 chapExoCorrec/3640 sacados/3640 chapExoCorrec/6097 sacados/6097
5. Convergences of monotone sequences E.6910 Consider the sequence u n defined by: u 0 = 5 ; u n +1 = 1 3 · u n + 4 for all n ∈ N 1 Using reasoning by recurrence, show that the sequence u n is increased by 7 . 2 Using reasoning by recurrence, show that the sequence u n is increasing. 3 Deduce that the sequence u n is convergent. E.3418 Consider the sequence u n n ∈ N ∗ defined by: u 1 = 2 5 ; u n +1 = 1 5 · u n + 2 5 for all n 1 1 Show that the sequence u n is increased by 1 . 2 Demonstrate that the sequence u n is increasing. 3 Justify that the sequence u n is convergent (the value of the limit is not asked) . E.3656 Consider the sequence u n defined by: u 1 = 2 5 ; u n +1 = 1 5 · u n + 2 5 for all n 1 1 Using reasoning by recurrence, show that the sequence ( u n ) is increased by 1 . 2 Demonstrate that ( u n ) is increasing. 3 Justify that the sequence ( u n ) is convergent. E.3649 The sequence u n is defined by: u 0 =2 ; u n +1 = 1 3 · u n + 23 27 pour tout n ∈ N 1 Using reasoning by recurrence, show that for any natural number n , we have : u n 23 18 2 Study the monotonicity of the sequence u n . 3 Establish the convergence of the sequence u n . E.5732 Consider the sequence u n de-fined by: u 0 = 1 ; u n +1 = 2 u n for any natural number n . 1 Show that, for any natural number n : 0 <u n 2 . 2 Determine the direction of variation of the sequence u n . 3 Show that the sequence u n is convergent. The value of its limit is not asked. E.5080 Let u n be the sequence defined for any non-zero natural number n by: u 1 = 1 2 ; u n +1 = n + 1 2 · n · u n pour tout n ∈ N 1 Calculate u 2 , u 3 and u 4 . 2 a Demonstrate that, for any non-zero natural number n , u n is strictly positive. b Show that the sequence u n is decreasing. c What can we deduce from this for the sequence u n ? 3 For any non-zero natural number n , we pose : v n = u n n a Show that the sequence v n is geometric. Its reason and first term v 1 will be specified. b Deduce that, for any non-zero natural number n : u n = n 2 n E.6102 Let u n be a sequence defined on the set of natural numbers N by: u 0 = 2 ; u n +1 = 1 5 · u n + 3 × 0 ; 5 n for all n ∈ N . 1 a Determine the value of the term u 1 . b Prove, by recurrence, that for any non-zero natural number n , we have : u n 15 4 × 0 ; 5 n 2 Deduce that, for any non-zero natural number n : u n +1 − u n 0 3 Prove that the sequence u n is convergent. 6. Divergence of monotone sequences E.3473 Consider the sequence u n n ∈ N defined by: u 0 = 1 ; u n +1 = 1 3 · u n + n − 2 for any n ∈ N . 1 Calculate u 1 , u 2 and u 3 . 2 a Demonstrate that for any natural integer n 4 : u n 0 . b Deduce that for any natural number n 5 : u n n − 3 c Deduce the limit of the sequence u n n ∈ N . E.3442 Consider the sequence u n n ∈ N defined by: u 0 = 2 u n +1 = u n + 2 n + 2 pour tout n ∈ N 1 Study the monotonicity of the sequence u n . 2 a Establish, by reasoning through recurrence, the fol-lowing inequality for any natural number n : u n > n 2 b Deduce the limit of the sequence u n . https://chingmath.fr chapExoCorrec/6910 sacados/6910 chapExoCorrec/3418 sacados/3418 Extrait de Asie Juin 2008 chapExoCorrec/3656 sacados/3656 Asie Juin 2008 chapExoCorrec/3649 sacados/3649 Metropoles et Reunion Septembre 2007 chapExoCorrec/5732 sacados/5732 chapExoCorrec/5080 sacados/5080 Extrait Antilles-Guyanes Juin 2012 chapExoCorrec/6102 sacados/6102 chapExoCorrec/3473 sacados/3473 chapExoCorrec/3442 sacados/3442
0√2∞∞√2∞xVariationdef 7. Sequences and variations of functions E.4234 Consider the function f defined on 0 ; + ∞ by: f ( x ) = 1 2 · x + 2 x We admit that the function f admits the following table of variations : Let u n be the sequence defined for any natural number n by: u 0 = 1 2 ; u n +1 = 1 2 · u n + 2 u n 1 Using reasoning by recurrence, show that for any non-zero natural number n , we have : u n 2 2 Show that for any x 2 : f ( x ) x . 3 Deduce that the sequence u n is decreasing from rank 1 . 4 Prove that the sequence u n converges. E.4224 Consider the function f defined on 0 ; + ∞ by: f ( x ) = ln x +1 + 1 2 · x 2 1 Study the direction of variation of the function f on the interval 0 ; + ∞ . 2 Consider the sequence u n defined on N by: u 0 = 1 ; u n +1 = f u n for all n ∈ N Show using reasoning by recurrence that, for any natural number n : u n 1 . E.3645 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = 6 − 5 x + 1 We admit that the function f is strictly increasing on 0 ; + ∞ . We note ¸ the unique positive real number verifying: f ( ¸ ) = ¸ où ¸ = 5+ 29 2 Consider the sequence u n defined by: u 0 =0 ; u n +1 = f ( u n )=6 − 5 u n +1 for all n ∈ N 1 Show that if x belongs to the interval 0 ; ¸ , then f ( x ) belongs to the interval 0 ; ¸ . 2 Demonstrate by recurrence that, for any natural number n , we have : 0 u n u n +1 ¸ 3 Establish that the sequence u n is convergent. E.10388 Consider the function f defined on R ∗ + by: f ( x ) = ln( x ) + 2 Consider the sequence u n n ∈ N defined by: u 0 = e ; u n +1 = f u n 1 Establish that for any integer n ∈ N , we have : u n u n +1 2 a Establish that for any integer n ∈ N , we have : u n 5 b Establish that the sequence u n is convergent. 8. Gendarme theorem E.3446 Consider the sequence u n n ∈ N verifying the relation: u 0 = 4 u n +1 = 1 2 · u n + 5 u n for any natural number n ∈ N 1 Using reasoning by recurrence, show that for any natural number n , we have : u n > 0 2 Using reasoning by recurrence, establish that for any nat-ural number n , we have : u n − 5 0 3 Using reasoning by recurrence, establish for any non-zero natural number n the following inequality: u n − 5 1 2 n 4 Deduce from the previous questions the convergence of the sequence u n and its limit. E.5034 Consider the sequence u n defined on N by the relation: u 0 = 2 ; u n +1 = 1 2 · u n + 2 u n pour tout n ∈ N 1 a Establish the following equality for any natural num-ber n : u n +1 − 2 = u n − 2 2 2 · u n b Deduce, ùsing reasoning by recurrence, that for any natural number n , we have : u n 2 2 a Deduce from the previous questions the framework: 0 u n +1 − 2 u n − 2 2 b Using reasoning by recurrence, establish the framing below for any natural number n : 0 u n − 2 u 0 − 2 (2 n ) 3 a Give an approximate value of ⏐ ⏐ u 0 − 2 ⏐ ⏐ to the near- https://chingmath.fr chapExoCorrec/4234 sacados/4234 0√2∞∞√2∞xVariationdef chapExoCorrec/4224 sacados/4224 Extrait de Liban Juin 2008 chapExoCorrec/3645 sacados/3645 sacados/10388 chapExoCorrec/3446 sacados/3446 chapExoCorrec/5034 sacados/5034
est millimetre. b Deduce the limit of the sequence u n . E.5569 Consider the sequence v n de-fined for any natural number n by: v 0 = 1 ; v n +1 = 9 6 − v n for all n ∈ N 1 Demonstrate by recurrence that, for any n ∈ N : 0 <v n < 3 2 a Demonstrate that, for any natural number n : v n +1 − v n = (3 − v n ) 2 6 − v n b Deduce that the sequence v n is increasing. 3 Demonstrate that the sequence v n is convergent. 9. Strong recurrence E.5814 Consider the sequence defined by the relations : u 0 =3 ; u 1 =8 ; u n +1 =5 · u n − 6 · u n − 1 for all 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 3 Determine the limit of the sequence u n . 10. A little further into the recurrence E.3650 1 Let n be a natural number greater than or equal to 1 . Show that : n +1 k =2 1 10 k = 1 90 · 1 − 1 10 n This means : 1 10 2 + 1 10 3 + · · · + 1 10 n +1 = 1 90 · 1 − 1 10 n 2 The suite v n is defined by: v n =1.2777 ··· 7 with n consecutive decimals equal to 7 . Thus : v 0 =1.2 ; v 1 =1.27 ; v 2 =1.277 . Using the a , demonstrate that the limit of the sequence v n is a rational number r (i.e. the quotient of two in-tegers) . . E.5815 Consider the sequence u n de-fined by u 0 = 1 2 and such that for any natural number n : u n +1 = 3 · u n 1+2 · u n 1 a Calculate u 1 and u 2 . b Demonstrate, by recurrence, that for any natural num-ber n : 0 <u n . 2 We admit that for any natural number n : u n < 1 . a Demonstrate that the sequence u n is increasing. b Show that the sequence u n converges. (the value of the limit is not asked) . E.10610 We define the sequence u n defined by: u 0 = ; u n +1 = u n + 2 n for any integer n ∈ N Establish, using reasoning by recurrence, the following iden-tity for any n ∈ N and for any k ∈{ 0 ; 1; ::: ; n } : u n +1 = u n − k + 2 n − k · 2 k +1 − 1 11. Courses E.5498 1 Demonstrate, using reasoning by recurrence, the prop-erty P n for any natural number n : P n : ˇfor any real number x such that x> 0 : (1 + x ) n 1 + n · x ı This relation is called Bernoulli’s inequality 2 Show that for any real number q such that q ∈ 1 ; + ∞ , we have the limit: lim n ↦→ + ∞ q n = + ∞ 12. Unclassified financial years E.3431 Consider the three sequences a n n ∈ N , b n n ∈ N , c n n ∈ N defined by: a 0 = 1 ; b 0 = c 0 = 0 a n +1 = 1 2 · b n pour tout n ∈ N b n +1 = 1 3 · a n pour tout n ∈ N a n + b n + c n = 1 pour tout n ∈ N https://chingmath.fr chapExoCorrec/5569 sacados/5569 Extrait Liban 2013 chapExoCorrec/5814 sacados/5814 chapExoCorrec/3650 sacados/3650 Extrait Metropole et Reunion Septembre 2007 chapExoCorrec/5815 sacados/5815 chapExoCorrec/10610 sacados/10610 chapExoCorrec/5498 sacados/5498 chapExoCorrec/3431 sacados/3431
1 Show that, for any natural number n : a n +2 = 1 6 · a n . 2 Deduce that, for any natural number p : a 2 p = 1 6 p et a 2 p +1 = 0 b 2 p = 0 et b 2 p +1 = 1 3 · 1 6 p 3 Show that : lim n ↦→ + ∞ a n =0 . It is assumed that lim n ↦→ + ∞ b n =0 . What is the limit of c n when n tends to + ∞ . E.10611 Consider the sequence u n defined by: u 0 = 5 ; u n +1 = u n + 2 n for all n ∈ N Establish, using reasoning by recurrence : u n = 2 n + 4 for all n ∈ N E.5524 Consider the sequence u n defined by: u 1 = 0.1 ; u n +1 = 1 5 u n + 3 5 for all n ∈ N ∗ 1 Show by recurrence that, for any non-zero natural num-ber n , we have : u n = 3 4 − 13 4 · 1 5 n 2 Determine the limit of the sequence u n when n tends to + ∞ . 3 For what values of the natural integer n does one have : 3 4 − u n < 10 − 7 E.5306 The sequences u n n ∈ N and v n n ∈ N with real terms are defined by: u 0 =5 ; u 1 =31 ; u n +2 =12 · u n +1 − 35 · u n pour tout n ∈ N v 0 = − 1 ; v 1 = − 11 ; v n +2 =12 · v n +1 − 35 · v n pour tout n ∈ N We define the sequences x n n ∈ N and y n n ∈ N by: x n = u n + v n ; y n = u n − v n for any n ∈ N 1 Calculate x 0 and x 1 . Using reasoning by recurrence, show that the sequence x n is a geometric sequence of reason 5 . 2 Similarly, show that the sequence y n is a geometric sequence. 3 Calculate x n and y n as a function of n ; derive the calcu-lation of u n and v n as a function of n . E.10543 We wish to study the sequence ( u n ) of first term u 0 =5 defined by the following recurrence rela-tion : u n +1 = 1 3 u n + 4 for all n ∈ N 1 Establish, using reasoning by recurrence, that for any n ∈ N , we have : u n = 6 − 1 3 n 2 Deduce the convergence of the sequence u n and its limit. 3 Determine from which value of n , the terms of the se-quence u n verify: u n 5.9 E.10616 Establish, using reasoning by recur-rence and for any n ∈ N ∗ , the identity: 1 2 + 2 2 + · · · + n 2 = n n + 1 2 n + 1 6 https://chingmath.fr chapExoCorrec/10611 sacados/10611 chapExoCorrec/5524 sacados/5524 chapExoCorrec/5306 sacados/5306 Extrait du Bac C - Amiens-Rouen Juin 1983 sacados/10543 chapExoCorrec/10616 sacados/10616