A what are the values of 10 n 1 a n and 16 n 4 a n 10

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(a) What are the values of 10 n = 1 a n and 16 n = 4 a n ? 10 n = 1 a n = 16 n = 4 a n = (b) What is the value of a 3 ? a 3 = (c) Find a general formula for a n . a n = (d) Find the sum n = 1 a n . n = 1 a n = Solution: Solution: (a) 10 n = 1 a n = S 10 = 5 - 5 10 2 = 4 . 95; 2
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16 n = 4 a n = ( a 1 + ... + a 16 ) - ( a 1 + a 2 + a 3 ) = S 16 - S 3 = 5 - 5 16 2 - 5 - 5 3 2 = 5 9 - 5 256 0 . 536024305555556 . (b) a 3 = ( a 1 + a 2 + a 3 ) - ( a 1 + a 2 ) = S 3 - S 2 = 5 - 5 3 2 - 5 - 5 2 2 = 5 4 - 5 9 0 . 694444444444444 . (c) Since a n = S n - S n - 1 , we have: a n = S n - S n - 1 = 5 - 5 n 2 - 5 - 5 ( n - 1 ) 2 = 5 ( n - 1 ) 2 - 5 n 2 = 5 ( n 2 - ( n - 1 ) 2 ) ( n ( n - 1 )) 2 = 5 ( n 2 - n 2 + 2 n - 1 ) ( n ( n - 1 )) 2 = 5 ( 2 n - 1 ) ( n ( n - 1 )) 2 . (d) The sum n = 1 a n is the limit of the partial sums S N . Hence; n = 1 a n = lim N S N = lim N 5 - 5 N 2 = 5 . Answer(s) submitted: (incorrect) Correct Answers: 4.95 0.536024 0.694444 5*(2*n-1)/([n*(n-1)]ˆ2) 5 11. (1 point) Let s k = k n = 1 n ( . 1 ) n Find s 7 . s 7 = Answer(s) submitted: (incorrect) Correct Answers: 0.1234567 12. (1 point) Find the sum of the first 8 terms of the geometric sequence if the first term is 5 and the common ratio is -2. Answer(s) submitted: (incorrect) Correct Answers: -425 13. (1 point) The following series are geometric series or a sum of two geometric series. Determine whether each series converges or not. For the series which converge, enter the sum of the series. For the series which diverges enter ”DIV” (without quotes). (a) n = 1 7 n 6 n = , (b) n = 2 1 2 n = , (c) n = 0 2 n 6 2 n + 1 = , (d) n = 5 6 n 7 n = , (e) n = 1 8 n 8 n + 4 = , (f) n = 1 6 n + 2 n 7 n = . Answer(s) submitted: (incorrect) Correct Answers: DIV 0.5 0.176470588235294 3.23865056226572 DIV 6.4 14. (1 point) Find the sum of the following infinite series. If it is divergent, type ”Diverges” or ”D”. n = 1 n 5 Sum: Answer(s) submitted: 3
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(incorrect) Correct Answers: Diverges 15. (1 point) Determine whether the series is convergent or divergent. If convergent, find the sum; if divergent, enter div . n = 1 arctan ( 18 n ) Answer: Answer(s) submitted: (incorrect) Correct Answers: div 16. (1 point) Determine whether the series is convergent or divergent. If convergent, find the sum; if divergent, enter div .
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