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M408L hwk10 solutions

# M408L hwk10 solutions - Pham Quoc Homework 10 Due Apr 4...

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Pham, Quoc – Homework 10 – Due: Apr 4 2007, 3:00 am – Inst: Eric Katerman 1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. 001 (part 1 oF 1) 10 points ±ind a Formula For the general term a n oF the sequence n 2 , 6 , 10 , 14 , ... o assuming that the pattern oF the frst Few terms continues. 1. a n = 3 n - 1 2. a n = 5 n - 3 3. a n = n + 4 4. a n = n + 3 5. a n = 4 n - 2 correct Explanation: In the sequence n 2 , 6 , 10 , 14 , ... o each term is larger than the preceding one by 4, so a n = a 1 + d ( n - 1) = 2 + 4( n - 1) . Consequently, a n = 4 n - 2 . keywords: 002 (part 1 oF 1) 10 points ±ind a Formula For the general term a n oF the sequence n 1 , - 2 5 , 4 25 , - 8 125 , ... o assuming that the pattern oF the frst Few terms continues. 1. a n = - 1 2 · n 2. a n = - 2 5 · n - 1 correct 3. a n = - 2 5 · n 4. a n = - 5 2 · n 5. a n = - 5 2 · n - 1 6. a n = - 1 2 · n - 1 Explanation: In the sequence n 1 , - 2 5 , 4 25 , - 8 125 , ... o each term is - 2 5 times the preceeding one, i.e. , a n = - 2 5 · a n - 1 . Consequently, a n = - 2 5 · n - 1 since a 1 = 1. keywords: sequence, exponential 003 (part 1 oF 1) 10 points Determine iF the sequence { a n } converges, and iF it does, fnd its limit when a n = 5 n 5 - 4 n 3 + 3 5 n 4 + n 2 + 1 . 1. limit = 1 2. the sequence diverges correct

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Pham, Quoc – Homework 10 – Due: Apr 4 2007, 3:00 am – Inst: Eric Katerman 2 3. limit = 0 4. limit = - 4 5. limit = 3 Explanation: After division by n 4 we see that a n = 5 n - 4 n + 3 n 4 5 + 1 n 2 + 1 n 4 . Now 4 n , 3 n 4 , 1 n 2 , 1 n 4 -→ 0 as n → ∞ ; in particular, the denominator converges and has limit 5 6 = 0. Thus by properties of limits { a n } diverges since the sequence { 5 n } diverges. keywords: 004 (part 1 of 1) 10 points Determine whether the sequence { a n } con- verges or diverges when a n = 10 n 2 2 n + 1 - 5 n 2 + 2 n + 1 , and if it does, Fnd its limit 1. limit = 5 2 correct 2. limit = 5 6 3. the sequence diverges 4. limit = 5 4 5. limit = 0 Explanation: After bringing the two terms to a common denominator we see that a n = 10 n 3 + 10 n 2 - (2 n + 1)
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M408L hwk10 solutions - Pham Quoc Homework 10 Due Apr 4...

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