Grade 11 / Sum of the terms of a sequence 68 exercises corrected

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ChingQuizz : 7 exercises available for Quizz assessment : 1. Reminders on powers E.7585 Express each of the calculations in the form a n où a is a non-zero real numberzero ( x ∈ R ∗ ) and n a relative integer ( n ∈ Z ) : a 2 5 × 2 7 b 2 8 2 − 3 c 5 5 5 12 d 3 5 × 3 2 3 4 e 3 2 5 f 3 2 5 3 4 × 5 20 E.7586 1 Establish each of the following equalities: a 3 9 + 2 × 3 9 = 3 10 b 5 6 + 2 2 × 5 6 = 5 7 2 Establish each of the following equalities: a 2 5 + 2 6 = 3 × 2 5 b 3 9 − 3 7 = 8 × 3 7 2. Introductory activity with Python E.7581 Since the day their daughter Aline was born, the parents have deposited the sum of 100 e per year in a Livret A passbook account in their child’s name. It is assumed that over the study period, the interest rate on the passbook remained constant at 1 % . As shown above have constructed the terms u 0 , u 1 , . . . , u 18 associated with the value, on the day of Aline’s 18 birthday, of each sum deposited by the parents. 1 a Give the values of the terms u 0 , u 1 and u 2 . b Give the value of u 18 , approximated to the nearest hundredth, representing the sum acquired by the 100 e deposited on the day of his birth. 2 To determine the sum disposing of the Livret A on the day he 18 years old, we will use programming software. a In the chosen software, enter the following algorithm: S ← 0 For i ranging from 0 to 5 u ← 3+2 × i S ← S+u End For b Justify that, at the end of its execution, the variable S contains the sum of the first 6 terms of the arithmetic sequence of first term 3 and reason 2 . c Modify this algorithm so that the variable S con-tains, at the end of the algorithm’s execution, the sum present in the A passbook on the day of Aline’s 18 birthday. 3 We note S 18 the sum of the first 19 terms of the sequence u n : S 18 = u 0 + u 1 + u 2 + · · · + u 18 Of the four proposals below, only one is correct. a S 18 = 100 × 1.01 18 b S 18 = 100 × 1 − 1.01 17 1 − 1.01 c S 18 = 100 × 1 − 1.01 18 1 − 1.01 d S 18 = 100 × 1 − 1.01 19 1 − 1.01 Using the approximate value obtained with the software, conjecture the correct expression of S 18 . https://chingmath.fr chapExoCorrec/7585 sacados/7585 chapExoCorrec/7586 sacados/7586 chapExoCorrec/7581 sacados/7581 Naissance100u181ans100u172ans100u1617ans100u118ansu0
E.7583 The problem of the Sissa chessboard [. . . ] is a mathemat-ical problem that can be expressed as follows : ˇA grain of wheat is placed on the first square of a chessboard. If we double the number of grains on each square from the previous square (one grain on the first square, two on the second, four on the third, etc.. . . ) , how many grains of rice do we get in total¾’ Source : Wikipedia As a reminder, a chessboard is made up of 64 white or black squares. So, after filling the first three squares, there are 7 grains of wheat on the chess-board. How many grains of wheat are needed to fill the chess-board? 1 How many grains of wheat will be placed in the 10 ème square? We want to approach the answer to this exercise using a pro-gramming language. 2 a In the chosen language, enter the following algo-rithm : S ← 0 For i ranging from 0 to 5 u ← 3+2 × i S ← S+u End For b Justify that at the end of its execution, variable S con-tains the sum of the first 6 terms of the arithmetic sequence with first term 3 and common difference 2 3 Adapt this algorithm so that, at the end of execution, the variable S has a value equal to the number of wheat grains on the board at the end of the game. Give the approximate value of the variable S at the end of the algorithm. 4 Among the four proposals below, only one : a S = 1 × 2 63 b S = 1 − 2 63 1 − 2 c S = 1 − 2 64 1 − 2 d S = 1 − 2 65 1 − 2 Using the software, give the scientific notation of these values, rounded to two digits in the decimal part of the mantissa. Conjecture the exact expression of S . 3. Introductory activity E.7582 The Sierpinski (1916) carpet, named after its Polish creator, is constructed by a succession of steps defined by: Each white square is subdivided into 9 identical squares by dividing its sides into three segments of equal length, and the central square is colored black Here are the first six steps of this construction : 1 For figures obtained in step 5 and following : How many black squares with sides 1 3 contains the fig-ure? How many black squares of side 1 9 contains the figure? How many black squares of side 1 27 contains the figure? 2 Using the programming software, determine the exact number S 4 of black squares present at step 4 ? 3 Using the calculator, determine the values of q and n so that : S 4 = 1 − q n 1 − q https://chingmath.fr chapExoCorrec/7583 sacados/7583 chapExoCorrec/7582 sacados/7582 Figure initiale Étape 0 Étape 1 Étape 2 Étape 3 Étape 4
E.7584 A globetrotter has bet to cover 5 000 km on foot. He can, fresh and ready, cover 50 km in a day, but every day fatigue builds up and so his performance decreases by 1 % every day. We note u 0 the distance covered on the first day of the race and generally u n the n ème day of the race. 1 a Give the value of the terms u 0 , u 1 , u 2 . b Determine the distance traveled on 30 ème race day rounded to the nearest meter. 2 To determine the distance covered after 45 racing days, we’ll use an automated spreadsheet : a Copy and complete the spreadsheet below up to col-umn AY . b What formula must be entered in cell B2 in order to be copied to the right and the cell range B2:AT2 represent the distances of the first 45 days of racing. c Give the approximate value, to the nearest metre, of the distance covered by the runner over the first 45 days of racing. 3 We note S 45 the sum of the first 45 terms of the sequence u n : S 45 = u 0 + u 1 + · · · + u 44 Of the four propositions below, only one is correct. a S 45 = 50 × 0.99 45 b S 45 = 50 × 1 − 0.99 44 1 − 0.99 c S 45 = 50 × 1 − 0.99 45 1 − 0.99 d S 45 = 50 × 1 − 0.99 46 1 − 0.99 Using the approximate value obtained with the software, conjecture the correct expression of S 45 . 4. A first approach to recurrence E.7554 Consider the sequence u n geomet-ric of first term 5 and reason 3 . Let S n be the sum of n +1 terms of the sequence u n : S n = u 0 + u 1 + · · · + u n 1 Determine the value of S 3 . 2 a We admit the equality S 6 = 5 2 · 3 7 − 1 . Establish : S 6 + u 7 = 5 2 · 3 8 − 1 b Using the previous result and approach, establish a simplified form of the sum S 8 3 Of the formulas below, expressing the sum S n as a func-tion of n , only one is correct. Which is it? a u 0 + u 1 + · · · + u n = 5 · 3 n +1 − 1 1 − 3 b u 0 + u 1 + · · · + u n = 5 · 1 − 3 n 1 − 3 c u 0 + u 1 + · · · + u n = 5 · 1 − 3 n +1 1 − 3 d u 0 + u 1 + · · · + u n = 5 · 3 n +1 − 1 1 − 3 E.7555 Consider the sequence u n of first term 3 and reason r . Let S n be the sum of n +1 terms of the sequence u n : S n = u 0 + u 1 + · · · + u n 1 Establish that the sum S 3 admits as expression : S 3 = 12 + 6 · r 2 a The formula : S 10 = 11 × 6 + 10 · r 2 Deduce the relationship : S 11 = 6 × 6 + 11 · r b Establish the following implication: S 11 = 6 × 6 + 11 · r = ⇒ S 12 = 13 × 6 + 12 · r 2 3 Of the formulas below, only one is correct: a S n = n · 2 · u 0 + n · r 2 a S n = ( n + 1) · 2 · u 0 + n · r 2 a S n = n · 2 · u 0 + ( n + 1) · r 2 a S n = ( n + 1) · 2 · u 0 + ( n + 1) · r 2 Which formula can be conjectured to be correct? 5. Number of terms in a sequence of terms E.6528 Consider a sequence u n n ∈ N . De-termine the number of terms in each of the sums below : a u 0 + u 1 + u 2 + u 3 + u 4 + u 5 + u 6 + u 7 + u 8 b u 5 + u 6 + u 7 + u 8 + u 9 + u 10 + u 11 + u 12 c u 11 + u 12 + u 13 + : : : + u 25 + u 26 d u 8 + u 9 + u 10 + · · · + u 31 + u 32 https://chingmath.fr chapExoCorrec/7584 sacados/7584 123ABCDEFGHJourdecourse1234567Distanceparcourue(enkm) chapExoCorrec/7554 sacados/7554 chapExoCorrec/7555 sacados/7555 chapExoCorrec/6528 sacados/6528
E.5124 Let u n n ∈ N be a numerical se-quence. For each question, give the number of terms making up the sum : a u 0 + u 1 + · · · + u 32 b u 5 + u 6 + · · · + u 15 c u 0 + u 1 + · · · + u n d u 5 + u 6 + · · · + u n E.10502 Let u n n ∈ N be a numerical se-quence. For each question, give the number of terms making up the sum : a u k + u k +1 + · · · + u 100 b u k + u k +1 + · · · + u n c u 0 + u 2 + · · · + u 88 d u 3 k + u 3 k +3 + · · · + u 99 e 64 k =0 u k f 16 k =5 u 2 k E.6529 Below are ˇ logical ı sequences of num-bers. Determine the number of terms in each of these sums : a 1 + 4 + 9 + 16 + : : : + 144 + 169 b 3 + 7 + 11 + 15 + : : : + 79 + 83 c 1 4 + 1 2 + 1 + 2 + : : : + 256 + 512 d 16 + 32 + 64 + : : : + 2 15 + 2 16 6. Introduction to the sum of the terms of an arithmetic sequence E.9653 1 We wish to determine the value of the sum : S = 1 + 2 + 3 + 4 + · · · + 9 a Complete the operations, operating column by column first : b From the previous operation, deduce the value of 2 × S . c Deduce the value of S . 2 We wish to determine the value of the sum : S = 1 + 2 + 3 + 4 + · · · + 100 a Complete the operations, operating column by column first : b From the previous posed operation, deduce the value of 2 × S . c Deduce the value of S . 3 Use a similar approach to the previous questions to de-termine the value of the sum S  defined by: S  = 1 + 5 + 9 + 13 + · · · + 81 E.6532 Consider a suite u n arithmetic of first term u 0 and reason r . 1 Express u 1 , u 2 and u 3 as a function of u 0 and r . 2 Express u n , u n − 1 and u n − 2 in terms of n , u 0 and r . 3 Justify the following equality: u 2 + u n − 2 = u 1 + u n − 1 = u 0 + u n 7. Arithmetic sequence: sum of first terms E.8171 Proposition: Let u n be the arithmetic sequence with first term u 0 and common difference r . Let S be the sum of the first n +1 terms of the sequence u n . We have : S = u 0 + u 1 + · · · + u n = n + 1 u 0 + u n 2 Consider the arithmetic sequence u n with first term 2 and common difference 2 . Determine the sum S of the first 100 terms of the sequence u n . E.9694 Consider the sequence u n arith-metic of first term 3 and reason 5 . Determine the sum of its 33 first terms. E.9715 Consider the sequence u n , defined for any n ∈ N , arithmetic with first term − 10 and reason 3 . Determine the value of the sum S defined by: S = u 0 + u 1 + · · · + u 84 E. 10361 Consider the sequence v n de-fined for any natural integer n by: v n = 4 + 3 · n Determine the sum S of the first 20 terms of the sequence v n . https://chingmath.fr chapExoCorrec/5124 sacados/5124 chapExoCorrec/10502 sacados/10502 chapExoCorrec/6529 sacados/6529 chapExoCorrec/9653 sacados/9653 1928374655647391c1c2c3c4c5c6c7 11002993984975966957948931001c1c2c3c4c5c6c7c8c9 chapExoCorrec/6532 sacados/6532 chapExoCorrec/8171 sacados/8171 chapExoCorrec/9694 sacados/9694 chapExoCorrec/9715 sacados/9715 chapExoCorrec/10361 sacados/10361
E.5860 Consider the sequence u n n ∈ N arith-metic with first term − 3 and reason 4 . 1 Give the expression of the term u n as a function of its rank n . 2 What is the rank of the term in the sequence u n with value 605 ? 3 Determine the value of the sum S defined by: S = u 0 + u 1 + · + u 100 E.7799 We wish to determine the simplified form of the quotient A defined by: A = 1 + 2 + 3 + 4 + 5 + 6 + · · · + 31 1 + 6 + 11 + 16 + 21 + · · · + 151 1 Consider the suite u n arithmetic with first term 1 and reason 1 . Determine the value of the sum S defined by: S = u 0 + u 1 + u 2 + · · · + u 30 2 Noting the decomposition 496=2 4 × 31 , determine the simplified form of the quotient A . 8. Arithmetic sequences: sum and recognition of sequences E.8464 Consider the sum S defined by: S = 1 + 3 + 5 + · · · + 101 We admit that the terms of the sum S are the first successive terms of an arithmetic sequence u n defined on N . 1 a Give the characteristic elements of the sequence u n . b Determine the rank of the term in the sequence with 101 as its value. 2 Deduce the value of the sum S . E.8499 Consider the sum S defined by: S = 2 3 + 1 + 4 3 + 5 3 + · · · + 10 We admit that the terms of the sum S are the first successive terms of an arithmetic sequence u n defined on N . 1 Give the characteristic elements of the sequence u n and determine the rank of the term with value 10 . 2 Deduce the value of the sum S E.8500 Consider the sum S defined by: S = 3 3 + 2 3 3 + 3 + · · · + 16 3 3 We admit that the terms of the sum S are the first successive terms of a sequence u n arithmetic defined on N . 1 a Give the characteristic elements of u n . b determine the rank of the term in the sequence with 16 3 3 as its value. 2 Deduce the value of the sum S . E.9695 The sum S , defined below, is the sum of consecutive terms of an arithmetic sequence : S = 7 + 10 + 13 + · · · + 340 Leaving traces of your approach, determine the value of the sum S . E.8501 Consider the sum S defined by: S = 1+2+101+102+201+202+301+302+ · · · +1501+1502 Determine the value of S . E.6548 The prince asks Sissou to start by placing one grain of rice on the first square, then three grains on the second square, then five grains on the third square and so on to fill the chessboard shown opposite. Determine the number of grains of rice Sissou will need to com-plete the chessboard. 9. Arithmetic sequences: sums and equations E.2419 Let u n n ∈ N be the arithmetic se-quence with first term 2 and reason r We’re interested in the sum S of the first 13 terms of u n : S = u 0 + u 1 + · · · + u 11 + u 12 Determine the value of r so that : S =65 E.2426 Let u n n ∈ N be the arithmetic se-quence of first term 5 and reason 2 . For any non-zero integer k ( k ∈ N ∗ ) , note : S k = u 0 + u 1 + · · · + u k 1 Given the integer k , how many terms comprise the sum S k ? 2 Determine the value of the integer k so that : S k = 10 605 https://chingmath.fr chapExoCorrec/5860 sacados/5860 chapExoCorrec/7799 sacados/7799 chapExoCorrec/8464 sacados/8464 chapExoCorrec/8499 sacados/8499 chapExoCorrec/8500 sacados/8500 chapExoCorrec/9695 sacados/9695 chapExoCorrec/8501 sacados/8501 chapExoCorrec/6548 sacados/6548 chapExoCorrec/2419 sacados/2419 chapExoCorrec/2426 sacados/2426
E.10398 Consider the sequence u n arithmetic of first term 5 and reason r . The sum of the 72 first terms has the value : u 0 + u 1 + u 2 + · · · + u 71 = 76 Determine the value of reason r . (we’ll leave the steps of his reasoning) E.10403 Consider the sequence u n , defined for n ∈ N , with first term 2 and common difference r . The sum of the first 71 terms is : u 0 + u 1 + u 2 + · · · + u 70 = 497 Determine the value of the common difference r . (We will leave out the steps of the reasoning) 10. Arithmetic sequences: general formula E.7644 Proposition: Let u n be an arithmetic sequence with first term u 0 and common ratio r . We have the property: Consider the arithmetic sequence u n with first term 3 and common difference 2 . Determine the value of the sum : S = u 12 + u 13 + · · · + u 34 E.10362 Consider the sequence v n defined for any natural number n ( n ∈ N ) by: v n =2 − 3 · n Determine the value of the sum : S = v 4 + v 5 + · · · + v 15 E.2430 1 Let u n n ∈ N be the arithmetic sequence of first term 2 and reason 1 4 . Determine the sum S defined by: S = u 11 + u 12 + · · · + u 25 2 Let v n n ∈ N be the arithmetic sequence of first term 12 and reason − 3 . Determine the sum S defined by: S = v 5 + v 6 + · · · + v 13 11. Arithmetic sequences: general formula and equations E.8465 Let u n n ∈ N be an arithmetic se-quence of first term 2 and reason r Consider the sum S of the sums of the terms of u n ranging from u 5 to u 20 : S = u 5 + u 6 + · · · + u 19 + u 20 Determine the value of the reason r in order to achieve : S =132 E.8466 Let u n n ∈ N be the arithmetic se-quence of first term 5 and reason 2 . Consider the sum S of successive terms of rank 14 to the term of rank k where k is a natural number strictly greater than 14 . That is,: S = u 14 + u 15 + ··· + u k Determine the value of k so that : S =320 12. Geometric sequence: sum of first terms E.8172 Proposition: Let u n be the geometric sequence with first term u 0 and common ratio q . Let S be the sum of the first n +1 terms of the sequence u n . We have : S = u 0 + u 1 + · · · + u n = u 0 · 1 − q n +1 1 − q Consider the geometric sequence u n with first term 2 and common ratio 2 . Determine the sum S of the first 100 terms of the sequence u n . E.10399 Consider the geometric sequence of first term 12 and reason 4 . Determine the sum of the first 100 terms of this sequence. Hint: we’ll give the simplified expression for this sum. E.7645 Consider the suite v n n ∈ N geomet-ric with first term 12 and reason 1 4 . 1 Give the expression of the term v n as a function of its rank n . 2 What is the rank of the term in the sequence v n with value 3 64 3 Determine a simplified expression for the sum S defined by: S = v 0 + v 1 + · · · + v 30 E.10363 Consider the sequence v n defined for any natural integer n by: v n = 5 2 n Determine the sum S of the first 20 terms of the sequence v n . https://chingmath.fr chapExoCorrec/10398 sacados/10398 chapExoCorrec/10403 sacados/10403 chapExoCorrec/7644 sacados/7644 ukuk···unn−k·ukun2Nombresde termesPremier termeDernier terme chapExoCorrec/10362 sacados/10362 chapExoCorrec/2430 sacados/2430 chapExoCorrec/8465 sacados/8465 chapExoCorrec/8466 sacados/8466 chapExoCorrec/8172 sacados/8172 chapExoCorrec/10399 sacados/10399 chapExoCorrec/7645 sacados/7645 chapExoCorrec/10363 sacados/10363
E.7806 Consider the sequence u n defined by: u 0 = 5 ; u n +1 = u n + 2 n for any n ∈ N 1 Give the values of the first 4 terms of the sequence. 2 Let S n be the sum of the n +1 first terms of the sequence u n : S n = u 0 + u 1 + u 2 + · · · + u n a Establish the following identity for any natural num-ber n : S n +1 = S n + 1 − 2 n +1 1 − 2 + 5 b Deduce that the terms of the sequence u n admit as expression in function of n : u n = 2 n + 4 13. Geometric sequence: sum of first terms and equation E.2420 Let u n n ∈ N be the geometric se-quence with first term 2 and reason 1 2 . Let k be a non-zero natural number ( k ∈ N ) , let S be the sum of the first k +1 terms of the sequence u n . That is : S = u 0 + u 1 + ··· + u k Determine the value of k so that : S =4 − 1 2 8 E.2427 Solve the equation : 1 − q 3 1 − q = 39 25 E.10402 Let u n n ∈ N be the geometric sequence with first term 1 484 and common ratio 3 . For an integer k strictly greater than 0 , let S be the sum of the successive terms of the sequence u n from term 0 to term k : S = u 0 + u 1 + · · · + u k Determine the value of the integer k satisfying : S =61 Hint: We will use the table of powers of 3 : 3 0 =1 ; 3 3 =27 ; 3 6 =729 ; 3 9 =19683 ; 3 12 =531441 3 1 =3 ; 3 4 =81 ; 3 7 =12187 ; 3 10 =59049 ; 3 13 =1594323 3 2 =9 ; 3 5 =243 ; 3 8 =6561 ; 3 11 =177147 ; 3 14 =4782969 14. Geometric sequences: recognizing the general term E.8468 Consider the sum S defined by: S = 27 + 9 + 3 + · · · + 1 81 We admit that the terms of this sum are the consecutive terms of a sequence u n geometric. 1 Give the characteristic elements of the suite u n . 2 Determine the rank of the term in the sequence u n whose value is 1 81 , then give the number of terms in the sum S . 3 Determine the value of S . E.8473 Consider the sum S below : S = 1 + 2 + 2 + 2 2 + · · · + 8 2 We admit that the terms of this sum are the consecutive terms of a sequence u n geometric. 1 Give the characteristics of the geometric sequence u n . 2 Determine the rank of the term in the sequence u n whose value is 8 2 . Give the number of terms in the sum S . 3 Deduce the value of S . E.8469 Consider the following numerical sum : S n = 4 + 2 + 1 + 1 2 + · · · + 1 2 n where n ∈ N We admit that the terms of this sum are the first terms of a suite u n geometric. 1 Give the characteristic elements of the suite u n . 2 Determine the rank of the term in the sequence u n worth 1 2 n . Give the number of terms in the sum S n . 3 a Determine the value of S n as a function of n . b Justify that, whatever the value of n , the sum S n is increased by 8 . E.8509 Establish that the integer 7 20 − 1 is a multiple of the integer 6 . E.2432 Let x be a real number other than 1 . 1 Express the following sum in terms of x : S = 1 + x + x 2 + x 3 + : : : + x n 2 Deduce a factorization of the polynomial 1 − x n +1 . https://chingmath.fr chapExoCorrec/7806 sacados/7806 chapExoCorrec/2420 sacados/2420 chapExoCorrec/2427 sacados/2427 chapExoCorrec/10402 sacados/10402 chapExoCorrec/8468 sacados/8468 chapExoCorrec/8473 sacados/8473 chapExoCorrec/8469 sacados/8469 chapExoCorrec/8509 sacados/8509 chapExoCorrec/2432 sacados/2432
E.8173 The prince asks Sissou to start by depositing one grain of rice on the first square, then two grains on the second square, then four grains on the third square and so on, multiplying by 2 the num-ber of grains deposited on the next square until the chessboard is fully completed. Determine the number of grains of rice Sissou will need to complete the chessboard. E.7246 Below are the first six ˇ flakes of Helge Von Koch ı representing one of the simplest fractals : Here’s the procedure assigned to each segment of the broken line to build the figure in the next step : Each segment is divided into three equal parts. On the segment in the middle of the segment, we con-struct an equilateral triangle. The segment in the middle of the segment is deleted 1 For any natural number n , let u n denote the number of segments making up the Helge Von Koch flake at step n . Conjecture a recurrence relationship between the terms of the sequence u n . 2 For any natural number n , let v n be the length of the broken line forming the Von Koch flake at step n . Conjecture a recurrence relationship between the terms of the sequence v n . 15. Geometric sequence: general formula E.7608 Proposition: Let u n n ∈ N be a geometric sequence with first term u 0 and common ratio q . We have the property: Consider the geometric sequence u n with first term 4 and common ratio 3 . Determine the value of the sum : S = u 10 + u 11 + · · · + u 19 E.10364 Consider the sequence v n whose rank term n , a natural number ( n ∈ N ) , is defined by: v n = 3 4 n Determine the value of the sum S : S = v 5 + v 6 + · · · + v 12 E.2431 Let u n n ∈ N be the geometric se- quence with first term 5 and reason 2 3 . Determine the value of the sum : S = u 10 + u 11 + · · · + u 21 E.9696 Consider the sequence un defined for any n ∈ N geometric of first term 2 4 × 3 5 and reason 1 3 . De-termine the sum of the 100 first terms of the sequence u n . E.8472 Consider the sequence u n n ∈ N geo-metric with first term 1 and reason 2 . We note S the value of the sum : S = u 2 + u 3 + u 12 + u 13 + u 22 + u 23 + u 32 + u 33 + · · · + u 82 + u 83 + u 92 + u 93 Determine the value of S . E.10605 Let v n n ∈ N be the geometric sequence of first term 12 and reason − 1 2 . Determine the value of the sum : S = v 7 + v 8 + · · · + v 12 16. Geometric sequences: general formula and equations E.8470 Let u n n ∈ N be the geometric se-quence with first term 2 and common ratio 1 2 . For an integer k strictly greater than 4 , let S be the sum of the successive terms of the sequence u n from term 4 to term k : S = u 4 + u 5 + · · · + u k Determine the value of the integer k satisfying : S = 127 512 Hint: we will use the table of powers of 2 : 2 0 =1 ; 2 3 =8 ; 2 6 =64 ; 2 9 =512 ; 2 12 =4096 2 1 =2 ; 2 4 =16 ; 2 7 =128 ; 2 10 =1024 ; 2 13 =8192 2 2 =4 ; 2 5 =32 ; 2 8 =256 ; 2 11 =2048 ; 2 14 =16384 https://chingmath.fr chapExoCorrec/8173 sacados/8173 chapExoCorrec/7246 sacados/7246 ABFigure no0 ABFigure no1 ABFigure no2 ABFigure no3 ABFigure no4 ABFigure no5 Etape0Etape1Etape2Etape3 chapExoCorrec/7608 sacados/7608 ukuk···unuk·1−qn−k1−qPremier termeNombre de termes chapExoCorrec/10364 sacados/10364 chapExoCorrec/2431 sacados/2431 chapExoCorrec/9696 sacados/9696 chapExoCorrec/8472 sacados/8472 chapExoCorrec/10605 sacados/10605 chapExoCorrec/8470 sacados/8470
E.10400 Let u n n ∈ N be the geometric sequence with first term 1 4356 and common ratio 3 . For an integer k strictly greater than 2 , let S be the sum of the successive terms of the sequence u n from term 2 to term k : S = u 2 + u 3 + · · · + u k Determine the value of the integer k satisfying : S =61 Hint: we will use the table of powers of 3 : 3 0 =1 ; 3 3 =27 ; 3 6 =729 ; 3 9 =19683 ; 3 12 =531441 3 1 =3 ; 3 4 =81 ; 3 7 =12187 ; 3 10 =59049 ; 3 13 =1594323 3 2 =9 ; 3 5 =243 ; 3 8 =6561 ; 3 11 =177147 ; 3 14 =4782969 17. A little further on E.5818 Let w n n ∈ N an arithmetic sequence with first term 1 and common difference 1 8 . Consider the fol-lowing sum : S 1 = w 0 + w 1 + · · · + w n Determine the value of n so that the sum S 1 has a value of 31 . (We will need to find the roots of the quadratic polynomial 2+ x 8 x +1 − 62 ) E.8423 Below is a rectangle ABCD verifying: AB = 2 cm ; AD = 1 cm Inscribed inside the rectangle are rectangles whose sides are parallel to the sides of the ABCD rectangle. Note A i , where i ∈ 1 ; 2 ; 3 ; 4 ; 5 1 Show that the sum A of the areas of the shaded domains is equal to the sum of the terms of a geometric sequence. 2 Justify that the sum A has value 2 − 1 16 . E.6549 We wish to determine the value of the following sum S : S = 9 + 15 + 27 + · · · + 3075 Note that this sum can be written as : S = 3 × 2 1 +3 + 3 × 2 2 +3 + 3 × 2 3 +3 + ··· + 3 × 2 10 +3 Determine the value of S All search traces, even if incomplete, will be taken into account in the evaluation . E.7660 Consider a function defined on R + whose representative curve is given below in a reference frame O ; I ; J : Furthermore, the set of points A n of the plane defined for any natural number n by their coordinates A n ( n ; 3 × 0.8 n ) belong to the curve C f . Any trace of research or reasoning, even if incomplete, will be taken into account and valued. 1 A plane domain is defined by considering the thirteen rectangles shown below : où the points A 0 , A 1 , . . . , A 12 form the vertices ˇen top to gaucheı of each of its rectangles. Determine the area of this domain. 2 A domain of the plane is defined by considering the thir-teen rectangles shown below : où the points A 1 , A 2 , . . . , A 13 form the vertices ˇen top to droiteı of each of its rectangles. Determine the area of this domain. https://chingmath.fr chapExoCorrec/10400 sacados/10400 chapExoCorrec/5818 sacados/5818 chapExoCorrec/8423 sacados/8423 ABCDA1A2A3A4A5 chapExoCorrec/6549 sacados/6549 chapExoCorrec/7660 sacados/7660 234567891011121314I23JO0A1A2A3A4A5A6A7A8A9A10A11A12A13A14A15ACf 234567891011121314I23JO 234567891011121314I23JO
E.7661 A ˇ snail exponentiel ı of parame-ter ¸ is a broken line whose ends A i segments form a sequence of points verifying the relations : For any natural number i : A i A i +1 = ¸ i For any natural number i , the triangle A 0 A i A i +1 is right-angled at A i . angle −−−→ A 0 A i ; −−−−−→ A 0 A i +1 is of positive measure. Below is the exponential snail of parameter 1.1 : In the rest of the exercise, the exponential snail has parameter 2 . Any trace of research or reasoning, even if incomplete, will be taken into account and valued. 1 Determine the length of the broken line A 0 A 1 A 2 : : : A 10 A 11 . 2 Determine the length of segment [ A 0 A 11 ] . E.10628 Let a n n ∈ N be a geometric sequence with first term 2 such that : a 0 + a 1 + a 2 + a 3 + a 4 + a 5 = 63 16 Determine the ratio of this sequence. (We will assume that the polynomial − 32 x 6 +63 x − 31 has roots 1 2 and 1 ) 18. Unclassified financial years E.6036 We have 800 small cubes. What is the height of the largest constructible pyramid following the model opposite? E.6648 Consider the two sequences a n and b n defined jointly by the two relations : a 0 = 0 a n +1 = 2 3 · a n + 1 3 · b n b 0 = 12 b n +1 = 1 4 · a n + 3 4 · b n ∀ n ∈ N 1 We define the sequence u n by the relation: u n = b n − a n a Establish that the sequence u n verifies the recurrent relation: u n +1 = 5 12 · u n b Give the nature of the sequence u n and its charac-teristic elements. c Give the sum S of the first 10 terms of the sequence u n . 2 We define the sequence v n by the relation: v n = 3 · a n + 4 · b n a Show that the sequence v n is constant. b Determine the sum S  of the first 10 terms of the se-quence v n . 3 Deduce the sum S defined by: S = b 0 + b 1 + · · · + b 9 E.9847 A globetrotter has bet to cover 5 000 km on foot. He can, fresh and ready, cover 50 km in a day, but every day, fatigue builds up and so his performance decreases by 1 % every day. We note u 1 the distance covered on the first day of the race and generally u n the n ème day of the race. 1 a Give the value of the terms u 1 , u 2 , u 3 . b Specify the nature of the sequence u n and its char-acteristic elements. c Will give distance covered on 30 ème day of race rounded to the nearest metre. 2 We note v n the sequence whose rank term n has the value of the total distance covered by the globe-trotter in the first n days a As a function of n , determine the expression for the term of rank n of the sequence v n . b Give the total distance, rounded to the nearest metre, covered by the globe-trotter over the first 30 days of the race. E.11576 Consider a sequence u n defined on N and let S n be the sum of its first n terms. We have the following table of values : n 0 1 2 3 4 5 S n 5 13 24 38 55 75 Determine the nature of the sequence u n and its character-istic elements. https://chingmath.fr chapExoCorrec/7661 sacados/7661 A1A2A3A4A5A6A7A8A9A10A11A0 chapExoCorrec/10628 sacados/10628 chapExoCorrec/6036 sacados/6036 chapExoCorrec/6648 sacados/6648 chapExoCorrec/9847 sacados/9847 sacados/11576