Grade 11 / Sequence generation 63 exercises (including 61 corrected)

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ChingQuizz : 3 exercises available for Quizz assessment : 1. Explicit generation mode E.7184 For each question, determine the first four terms of the sequence u n n ∈ N : a u n = n + 1 n + 2 b u n = n 2 + n + 1 c u n = n − 2 n + 1 E.9484 For each question, determine the first five terms of the sequence u n defined by: a u n = 3 · n 2 + n + 2 n + 2 for all n ∈ N b u n = 2 n 2 + n + 5 n + 1 for any n ∈ N E.7185 Consider the sequence u n n ∈ N whose rank term n is given by the relation: u n = − 7 4 · 1 − ( − 1) n + 3 1 Determine the first five terms of this sequence. 2 What can be said about the value of the terms in the sequence u n ? E.8486 Consider the sequence v n n ∈ N de-fined by the explicit formula : v n = 2 n 2 − 3 n + 2 for any natural number n . We wish to study the difference between two consecutive terms of the sequence ( v n ) : 1 Give the expression for the term v n +1 as a function of n . 2 Study the value of v n +1 − v n as a function of n . 2. Recurrence generation mode E.2377 We define the recurrence sequence u n n ∈ N by the relation: u 0 =5 ; u n +1 = 2 · u n − 1 for all n ∈ N Determine the first five terms of the sequence ( u n ) . E.9510 Consider the sequence v n n ∈ R de-fined by: v n +1 = 1 − v n 1 + v n ; v 0 = 3 1 Determine the first five terms of the sequence v n . 2 What do we notice? E.9509 For each question, determine the first four terms of the sequence u n n ∈ N : a u n +1 = 2 · u n − 2 ; u 0 = 3 b u n +1 = 2 · u n − 2 ; u 0 = 1 c u n +1 = u n − 2 u n + 1 ; u 0 = 2 E.5041 Justify that, in each question, the information below does not define sequences : a u 0 = 5 ; u n +1 = 2 · u n − 3 for any n ∈ N ∗ b u 0 = 1 ; u 1 = 4 ; u n +1 = u n − 3 for any n ∈ N c u 0 = 3 ; u n = 2 · u n − 1 − 2 for all n ∈ N d u 0 = − 1 ; u n = u n − 1 − 2 u n − 1 + 1 pour tout n ∈ N ∗ 3. 2nd-order recurrence generation mode E.9513 Consider the sequence u n defined for any n ∈ N by the relations : u 0 = 2 ; u 1 = 3 ; u n +1 = u n + 2 · u n − 1 for all n ∈ N ∗ Determine the first four terms of the sequence u n . E.5857 Consider the sequence u n defined by: u 0 = 3 ; u 1 = 1 u n +2 = 2 · u n +1 + u n pour tout n ∈ N Give the first five terms of the sequence u n . E.10358 Consider the sequence u n defined for any natural number n by: u 0 = 1 ; u 1 = − 2 ; u n +2 = 2 · u n +1 − u n + 2 u n +1 + u n − 2 1 Determine the first 4 terms of the suite u n . 2 What can be deduced from the sequence u n . E.10359 Consider the sequence u n defined for any natural number n by: u 0 = 2 ; u 1 = − 2 ; u n +2 = u n +1 − u n + 2 2 · u n +1 + 2 · u n + 2 Show that the sequence u n is periodic of period 2 from the term of rank 4 . https://chingmath.fr chapExoCorrec/7184 sacados/7184 chapExoCorrec/9484 sacados/9484 chapExoCorrec/7185 sacados/7185 chapExoCorrec/8486 sacados/8486 chapExoCorrec/2377 sacados/2377 chapExoCorrec/9510 sacados/9510 chapExoCorrec/9509 sacados/9509 chapExoCorrec/5041 sacados/5041 chapExoCorrec/9513 sacados/9513 chapExoCorrec/5857 sacados/5857 chapExoCorrec/10358 sacados/10358 chapExoCorrec/10359 sacados/10359
Etape no0 Etape no1 Etape no2 4. Other generation modes E.9512 Consider the sequence u n defined for any n ∈ N by the relations : u 0 = − 1 ; u n +1 = u n + n − 2 for all n ∈ N Determine the first four terms of the sequence u n . E.8405 Consider the sequence u n defined for any n ∈ N by: u 0 = 2 ; u n +1 = 3 · u n − 2 · n + 1 Determine the first four terms of the sequence u n . E.9515 Consider the sequence v n defined by: v 0 = − 3 ; v n +1 = n − 2 · v n for all n ∈ N Give the first four terms of the sequence v n . E.9539 Consider the sequence u n defined for any n ∈ N by: u 0 = 5 ; u n +1 = u n − 4 · n + 3 Determine the first four terms of the sequence u n . E.5104 Consider the sequence u n defined for any n ∈ N by the relations : u 0 = 2 ; u n +1 = 1 2 · u n + 3 for all n ∈ N Determine the first four terms of the sequence u n . E.8045 Consider the sequence u n defined for any n ∈ N by: u 0 = 1 ; u n +1 = 4 · u n 3 · n − 2 Determine the first five terms of the sequence u n E.8404 Consider the sequence v n defined for any n ∈ N by: v 0 = 1 ; v n +1 = 3 · v n 2 · n − 3 Determine the first six terms of the sequence v n E.5134 For each question, determine the first five terms of the sequence u n defined by: v 0 = 1 ; v n +1 = − 4 · v n 3 · n − 4 pour tout n ∈ N E.9511 We define the recurrence sequence v n n ∈ N ∗ by the relation: v 1 = − 2 ; v n +1 = 1 − v n n for any n ∈ N ∗ Determine the first five terms of the sequence ( v n ) . E.4628 Consider the construction of a figure in successive steps : In step 0 , the figure consists of a square with side 4 . A series of steps is constructed by adding a square whose side measures half the previously added square. Here are the first three steps in constructing this figure : We note u n the total area of the figure constructed in step n e . Thus, the sequence u n is defined for any natural num-ber n and we have : u 0 = 16 1 Justify that the sequence u n verifies the recurrence re-lation: u n +1 = u n + 4 2 2 n 2 We admit the existence of two real numbers ¸ and ˛ such that the sequence u n admits as explicit expression : u n = ¸ + ˛ · 1 4 n Conjecture the values of ¸ and ˛ E.7306 Consider the sequence u n defined by: u 0 = 1 ; u n +1 = n + 2 · u n + 1 n + 1 pour tout n ∈ N 1 Determine the first four terms of the sequence u n . 2 Conjecture the nature of the sequence u n , justifying your approach. 5. Using auxiliary suites E.3020 Let u n n ∈ N be the sequence defined by the relation: u n = 7 × 4 n − 2 × 3 n 1 Show that the sequence ( u n ) verifies the following rela-tionship : u n +2 = 7 · u n +1 − 12 · u n . 2 Consider the sequence v n n ∈ N defined by the relation: v n = u n +1 − 3 · u n Show that the sequence ( v n ) is a geometric sequence. We will give the first term and the reason. 6. Jointly defined sequences https://chingmath.fr chapExoCorrec/9512 sacados/9512 chapExoCorrec/8405 sacados/8405 chapExoCorrec/9515 sacados/9515 chapExoCorrec/9539 sacados/9539 chapExoCorrec/5104 sacados/5104 chapExoCorrec/8045 sacados/8045 chapExoCorrec/8404 sacados/8404 chapExoCorrec/5134 sacados/5134 chapExoCorrec/9511 sacados/9511 chapExoCorrec/4628 sacados/4628 Etape no0 Etape no1 Etape no2 chapExoCorrec/7306 sacados/7306 chapExoCorrec/3020 sacados/3020
1234567ABCDnunvntn01234537153163255581011111064-8-16032 E.9514 Consider the two sequences u n and v n defined by the relationships : u 0 = − 2 ; v 0 =1 ; u n +1 = u n + v n − 3 v n +1 = 2 · u n − v n + 1 for all n ∈ N Determine the first four terms of these two sequences. E.9541 Consider the two sequences u n and v n defined for any natural number n by: u 0 = 2 ; v 0 = 3 ; u n +1 = 2 · u n + v n 3 v n +1 = u n + 5 · v n 6 1 Determine the first three terms of each of the sequences u n and v n . 2 Consider the sequence w n defined for any natural num-ber n by: w n = u n − v n Demonstrate that the sequence w n is a geometric se-quence whose characteristic elements will be specified. E.10396 Consider the two sequences u n and v n defined on N by: u 0 = 10 ; v 0 = 7 ; u n +1 = u n − 4 · v n 3 v n +1 = − 5 · u n + 2 · v n 3 1 Determine the first three terms of each of these two se-quences. 2 We define the sequence w n , for any natural number n , by: w n = u n − v n a Establish that the sequence w n is a geometric se-quence. Specify its reason. b Knowing that u 19 =699 042 and leaving the steps of your reasoning, determine the value of v 19 . E.10397 Consider the sequences u n and v n defined for all n ∈ N by: u 0 =5 ; v 0 =2 ; u n +1 = 4 · u n − 3 · v n v n +1 = 2 · u n − v n for all n ∈ N 1 We define the sequence w n defined for all integers n ∈ N : w n = u n − v n a Establish that the sequence w n is a geometric se-quence with common ratio 2 , whose first term we will specify. b Give the explicit expression of the terms of the se-quence w n in terms of n . 2 Knowing that v 22 =25 165 820 , determine the value of the term u 22 . E.10454 Consider the sequences u n and v n defined on N by: u 0 = 2 u n +1 = 0.1 · u n + 0.4 · v n u n + v n = 5 Justify that these two sequences verify the conditions below : u 0 = 2 ; v 0 = 3 u n +1 = 0.1 · u n + 0.4 · v n v n +1 = 0.9 · u n + 0.6 · v n 7. All generation modes E.6645 Consider three sequences u n , v n and t n whose first terms have been given in the spreadsheet below : 1 Verify that the formulas below are verified by the values in the table : B 5=2 ∗ B 4+1 C 3= C 2 − A 2+3 D 6= D 5 − 2 ∗ D 4 2 Use these formulas to derive the recurrence formula defin-ing each of the terms in these sequences. E.6522 Consider the following sequence of numbers : a 4 ; 7 ; 10 ; 13 ; 16 ; 19 ; 22 . . . b 1 ; − 2 ; 4 ; − 8 ; 16 ; − 32 ; 64 . . . c 2 ; 2 ; 3 ; 5 ; 8 ; 12 ; 17 . . . d 0 ; 1 ; 4 ; 9 ; 16 ; 25 ; 36 . . . e 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 . . . f 1 ; 2 ; 1 ; 2 ; 1 ; 2 ; 1 . . . Associate with each of these sequences a relation below which allows us to obtain a term according to its predecessors : 1 u n + u n +1 = u n +2 2 2 u n = u n +1 3 u n + n = u n +1 4 − 2 × u n = u n +1 5 u n + 3 = u n +1 6 u n = n 2 https://chingmath.fr chapExoCorrec/9514 sacados/9514 chapExoCorrec/9541 sacados/9541 chapExoCorrec/10396 sacados/10396 chapExoCorrec/10397 sacados/10397 fichierPlus/10397/ chapExoCorrec/10454 sacados/10454 chapExoCorrec/6645 sacados/6645 1234567ABCDnunvntn01234537153163255581011111064-8-16032 chapExoCorrec/6522 sacados/6522
u0u1u2 Une plancheu1u2u3 1erétape2ièmeétape3ièmeétape 1erétape2ièmeétape3ièmeétape E.7305 Consider a sequence u n whose first five terms are known : u 0 =0 ; u 1 =11 ; u 2 =20 ; u 3 =27 ; u 4 =32 Which of the sequence expressions below yield these same first five terms? a u 0 = 0 u n +1 = u n + n + 11 pour tout n ∈ N b u 0 = 0 u n +1 = − u n + 3 n + 11 pour tout n ∈ N c u 0 = 0 u n +1 = u n − 2 n + 11 pour tout n ∈ N d u n = 13 · n − 2 · n 2 e u n = − n 2 + 12 · n f u n = 2 · n 2 + 9 · n 8. Going deeper: a little further E.2986 Consider the construction of a house of cards : Consider the sequence u n n ∈ N denoting the number of cards used in building the castle at step n . 1 Determine the first four terms of the sequence u n . 2 For any natural number n , determine an expression for the term u n +1 as a function of the preceding term u n and the rank n . 3 At what stage of construction can we arrive with two 72-card decks? E.5858 An object is built successively as shown in the diagram below : For any non-zero natural number n , note u n the number of boards required to construct the figure at step n . Give a recurrence relation characterizing the sequence u n . E.6132 Consider the following construc-tions : Note u n the numerical sequence defined on N ∗ où u n rep-resents the number of matches required to build the n ième step. 1 Determine a recurrence relationship between a term in the sequence u n and its predecessor. 2 Consider the sequence v n defined by the relation: v n = 2 · n 2 + 2 · n for all n ∈ N Using a spreadsheet, determine the 10 first terms of the sequences u n and v n . E.7245 Consider the construction of a house of cards : We note u n the number of cards required to build the house of cards at step n . This defines a sequence u n defined on N ∗ . Conjecture a recurrence relation on the terms of the sequence u n . https://chingmath.fr chapExoCorrec/7305 sacados/7305 chapExoCorrec/2986 sacados/2986 u0u1u2 chapExoCorrec/5858 sacados/5858 Une plancheu1u2u3 chapExoCorrec/6132 sacados/6132 1erétape2ièmeétape3ièmeétape chapExoCorrec/7245 sacados/7245 1erétape2ièmeétape3ièmeétape
1erétape2ièmeétape3ièmeétape 1erétape2ièmeétape3ièmeétape Etape 1 Etape 2 Etape 3 Etape 4 E.8047 Consider the following step-by-step construction of equilateral triangles using matchsticks : For any non-zero natural number n , note u n the number of matches required to construct the figure in step n . Thus, we have : u 1 =3 1 Which of the relationships below verifies the terms of the sequence u n : a u n +1 = 3 · u n + 3 b u n +1 = u n + 3 · n + 3 c u n +1 = u n + 6 · n d u n +1 = u n − 3 · n + 9 2 Which of the following relationships verifies the terms of the sequence u n : a u n = 3 2 · n 2 + 3 2 · n b u n = n 2 + 2 · n c u n = 3 2 · n 2 − 1 2 · n + 1 d u n = n 2 + 3 2 · n + 1 2 3 Give the value of the term u 6 . E.7244 Consider the following construc-tions : Note u n the numerical sequence defined on N ∗ où u n rep-resents the number of matches required to build the n ième step. Conjecture a recurrence relationship between a term in the sequence u n and its predecessor. E.7310 Below are the recurring stages in the construction of a geometric figure In each step, each segment of the figure is divided into 3 equal parts and on the middle segment, a square is constructed from which the middle segment is erased : Knowing that the square from step 1 has sides that measure 1 , determine the perimeter of the figure obtained in step 4 . The exact value and the value to the nearest hundredth will be given. https://chingmath.fr chapExoCorrec/8047 sacados/8047 1erétape2ièmeétape3ièmeétape chapExoCorrec/7244 sacados/7244 1erétape2ièmeétape3ièmeétape chapExoCorrec/7310 sacados/7310 Etape 1 Etape 2 Etape 3 Etape 4
Etape 1Etape 2Etape 3Etape 4 Etape 1Etape 2Etape 3Etape 4 Etape 1Etape 2Etape 3Etape 4 Etape 1Etape 2Etape 3Etape 4 E.11578 We will consider constructing a figure step by step below : where we gradually surround a square with sides of length 2 with squares with sides of length 1 . Let u n be the number of tiles used in step n . 1 Complete the table below : n 1 2 3 4 5 u n 2 Determine the values of a;b ∈ R that satisfy the relation: u n +1 = u n + a · n + b The figure below shows the elements added from one step to the next : Let v n be the number of tiles added to obtain the figure in step n . 2 a Complete the table below : n 1 2 3 4 5 v n b What is the nature of the sequence v n ? Give its characteristic elements. 3 We note that : u 2 = v 1 + v 2 ; u 3 = v 1 + v 2 + v 3 ; . . . a How can we describe the terms of the sequence u n in terms of the terms of the sequence v n ? b Establish that for all n ∈ N ∗ , we have : u n = 4 · n 2 + 8 · n E.11583 We will consider constructing a figure step by step below : where we gradually surround a square with sides of length 2 with squares with sides of length 1 . Let u n be the number of tiles used in step n . 1 Complete the table below : n 1 2 3 4 5 u n 2 Determine the values of a;b ∈ R that satisfy the relation: u n +1 = u n + a · n + b The figure below shows the elements added from one step to the next : Let v n be the number of tiles added to obtain the figure in step n . 2 a Complete the table below : n 1 2 3 4 5 v n b What is the nature of the sequence v n ? Give its characteristic elements. 3 We note that : u 2 = v 1 + v 2 ; u 3 = v 1 + v 2 + v 3 ; . . . a How can we describe the terms of the sequence u n in terms of the terms of the sequence v n ? b Establish that for all n ∈ N ∗ , we have : u n = 3 · n 2 + 3 · n 9. ICT activity E.7556 Consider the sequence u n defined by: u 0 = 1 ; u n +1 = 2 · u n + 3 n for any n ∈ N . 1 a Check the value of the following two terms : u 1 = 3 ; u 2 = 9 b Determine the value of the term of rank 3 of the se-quence u n . 2 a Complete the algorithm below so that the variable u successively takes the first 20 terms of the sequence u n u ← 1 For i ranging from 0 to ... u ← ... End For b Enter this algorithm into AlgoBox so that it displays the first 20 terms of the sequence u n . What con-jecture can be made about the nature of the sequence https://chingmath.fr sacados/11578 Etape 1Etape 2Etape 3Etape 4 Etape 1Etape 2Etape 3Etape 4 sacados/11583 Etape 1Etape 2Etape 3Etape 4 Etape 1Etape 2Etape 3Etape 4 chapExoCorrec/7556 sacados/7556
u n ? E.7557 Consider the sequence u n defined by: u 0 = 3 ; u n +1 = 9 × 2 n − u n 1 a Check the value of the following two terms : u 1 = 6 ; u 2 = 12 b Determine the value of the term of rank 3 of the se-quence u n . 2 a Using a spreadsheet, generate the first 20 terms of this sequence. b What conjecture can be made about the nature of the sequence u n E.7558 Consider the sequence u n defined by: u 0 = 1 ; u n +1 = n + 2 · u n + 1 n + 1 pour tout n ∈ N 1 a Check the value of the following two terms : u 1 = 3 ; u 2 = 5 b Determine the value of the term of rank 3 of the se-quence u n . 2 a Using a spreadsheet, generate the first 20 terms of this sequence. b What conjecture can be made about the nature of the sequence u n E.7559 Consider the sequence u n n ∈ N de-fined by the recurrence relation and verifying the conditions : u 0 = 5 ; u 1 = 11 ; u n +2 = 2 · u n +1 − u n for all n ∈ N 1 a Check the value of the following two terms : u 2 = 17 ; u 3 = 23 b Determine the value of the term of rank 4 of the se-quence u n . 2 a Complete the following algorithm so that the vari-able a takes during the execution of the algorithm the first 20 terms of the sequence u n : a ← 5 b ← a a ← 11 For i ranging from 2 to ... c ← a a ← ... b ← c End For b Enter this algorithm into AlgoBox so that it displays the first 20 terms of the sequence u n . What con-jecture can be made about the nature of the sequence u n ? E.7285 Consider the sequence u n geomet-ric with first term 2 and reason 2 : 1 Enter the algorithm below. n ← 0 u ← 2 As long as u<1000 u ← 2 × u n ← n + 1 End As long as Interpret the value of variable n at the end of the algo-rithm execution. 2 Modify the algorithm to know the rank of the first term greater than 5000 . E.8357 Consider the following algorithm: a ← 2 For i ranging from 0 to 5 a ← a + 3 End 1 In order to find out the value of the variable a at the end of the execution of this algorithm, enter this algorithm in the Python language : a=2; for i in range(0.6): a=a+3; print(a) 2 Which of the following sequences has been implemented in the previous algorithm? a u 0 = 3 u n +1 = u n + 2 b u 0 = 3 u n +1 = 2 × u n c u 0 = 2 u n +1 = u n + 3 d u 0 = 2 u n +1 = 3 × u n E.8358 Consider the sequence u n geomet-ric with first term 4 and reason 2 . 1 Which of the following algorithms displays the term of rank 8 of the sequence u n : a a ← 4 For i ranging from 0 to 8 a ← A × 2 End For Display a b a ← 4 For i ranging from 1 to 8 a ← A × 2 End For Display a c a ← 2 For i ranging from 0 to 8 a ← A × 4 End For Display a d a ← 2 For i ranging from 1 to 8 a ← A × 4 End For Display a 2 Modify the algorithm to obtain the value of the term u 12 https://chingmath.fr chapExoCorrec/7557 sacados/7557 chapExoCorrec/7558 sacados/7558 chapExoCorrec/7559 sacados/7559 chapExoCorrec/7285 sacados/7285 chapExoCorrec/8357 sacados/8357 chapExoCorrec/8358 sacados/8358
E.5092 Consider the following algorithm: a ← − 1 For i from 0 to 4 a ← a × 2 − i+1 End For 1 Give the different values taken by the variable a during a step-by-step execution of this algorithm. 2 Give the expression of a sequence u n whose first five terms are the different values taken by the variable a during the execution of this algorithm. E.5091 Consider the following algorithm: a ← 2 For i ranging from 0 to 5 a ← A × 2 End For 1 During its step-by-step execution, indicate the different values taken by the variable a 2 Among the chosen exrpressions that elle (s) can be the expression of a sequence u n so that its first six terms are the values taken by the variable a when executing the previous algorithm: a u n = 2 · n; ∀ n ∈ N b u n = 2 n ; ∀ n ∈ N c u n = 2 n +1 ; ∀ n ∈ N d u 0 = 2 u n +1 = 2 · u n ; ∀ n ∈ N e u 0 = 2 u n = 2 · u n +1 ; ∀ n ∈ N f u 0 = 2 u n = 2 · u n − 1 ; ∀ n ∈ N ∗ E.5090 Consider the following algo-rithm : For i from 0 to 5 a ← i × (i − 1) End For 1 When executing this algorithm step by step, give the val-ues taken by the variable a . 2 Give the expression of a sequence u n whose first six terms are the values displayed by the algorithm. E.7190 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 10. Arithmetic and geometric sequences E.9540 Consider the sequence v n defined for any n ∈ N by: v 0 = 2 ; v 1 = 3 ; v n +2 = v n +1 + 2 · v n − 3 Justify that the sequence v n is not an arithmetic sequence. E.10360 Consider the sequence u n defined for any n ∈ N by: u 0 = 4 ; u n +1 = 3 · u n − 4 · n − 4 Justify that the sequence u n is not a geometric sequence. E.9542 Consider the sequence u n defined for any n ∈ N by: u 0 = 4 ; u n +1 = 2 · u n − 3 · n − 1 1 a Determine the first four terms of the sequence u n . b What conjecture can be made for the nature of the sequence u n . 2 Consider the suite v n n ∈ N arithmetic of first term 4 and reason 3 . a Express the expression v n +1 − 2 · v n as a function of n . b Show that the sequences u n and v n are equal. E.9543 Consider the two sequences u n and v n defined for any natural number n by: u 0 = 3 ; v 0 = − 1 ; u n +1 = 6 · u n + 3 · v n 6 v n +1 = 8 · u n − 2 · v n 4 1 Determine the first three terms of each of the sequences u n and v n . 2 Consider the sequence w n defined for any natural num-ber n by: w n = u n − v n Demonstrate that the sequence w n is a geometric se-quence whose characteristic elements will be specified. 11. Unclassified financial years https://chingmath.fr chapExoCorrec/5092 sacados/5092 chapExoCorrec/5091 sacados/5091 chapExoCorrec/5090 sacados/5090 chapExoCorrec/7190 sacados/7190 chapExoCorrec/9540 sacados/9540 chapExoCorrec/10360 sacados/10360 chapExoCorrec/9542 sacados/9542 chapExoCorrec/9543 sacados/9543
E.8046 Consider the sequence u n defined on N by the relation: u 0 =1 ; u n +1 =3 · u n − 6 · n +1 for any n ∈ N 1 Determine the values of the first four terms. 2 What conjecture can be made about the nature of the sequence and its characteristic elements. E.5119 1 Consider the sequence u n defined by: u 0 = 1 ; u n +1 = 2 · u n + 3 n for any n ∈ N . a Determine the first five terms of u n . b What conjecture can be made about the nature of u n 2 Show that the geometric sequence v n with first term 1 and reason 3 verifies the relation: v n +1 = 2 · v n + 3 n . E.7304 Consider the sequence u n defined by: u 0 = 3 ; u n +1 = 9 × 2 n − u n for all n ∈ N 1 Determine the value of the first four terms of the se-quence u n . 2 Conjecture the nature of the sequence u n , justifying your approach. E.5173 Consider the two sequences a n and b n jointly defined by the relations : a 0 = 0.40 b 0 = 0.41 ; a n +1 = 0.6 a n + 0.3 b n b n +1 = 0.3 a n + 0.6 b n for all n ∈ N 1 Give the exact value of the first three terms of each of the sequences a n and b n . 2 We define the two suites u n and v n defined on N by: u n = a n + b n ; v n = b n − a n for any n ∈ N . a Show that the sequence u n is a geometric sequence of reason 0.9 . The first term will also be specified. b Show that the sequence v n is a geometric sequence whose characteristic elements will be specified. E.10225 1 Consider the sequence u n n ∈ N defined for any n ∈ N : u 0 = 0 ; u n +1 = u n + 2 · n + 2 Determine the first four terms of the sequence u n . 2 Consider the sequence v n n ∈ N defined, for n ∈ N , by: v n = n · n + 1 a Determine the first four terms of the sequence v n . b Expand and reduce the expression v n +1 − v n . c Deduce the equality of the sequences u n and v n . E.10324 Consider the sequence u n defined, for any n ∈ N , we have : u 0 = 2 ; u n +1 = u n + 3 · n 2 + 3 · n − 2 1 Expand, reduce and then factor the expression : n + 2 n − 1 2 + 3 · n 2 + 3 · n − 2 2 Deduce that the sequence u n admits as explicit form : u n = n + 2 n − 1 2 E.10617 Consider the sequence u n defined on N ∗ by the relation: u n = n n + 1 2 n + 1 6 Establish that the sequence u n satisfies the recurrence rela-tion : u n +1 = u n + n + 1 2 Note: we have just established that for any integer n ∈ N ∗ , we have : 1 2 + 2 2 + · · · + n 2 = n n + 1 2 n + 1 6 https://chingmath.fr chapExoCorrec/8046 sacados/8046 chapExoCorrec/5119 sacados/5119 chapExoCorrec/7304 sacados/7304 chapExoCorrec/5173 sacados/5173 chapExoCorrec/10225 sacados/10225 chapExoCorrec/10324 sacados/10324 chapExoCorrec/10617 sacados/10617