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Gilbert_Hmwk03sol

# Gilbert_Hmwk03sol - moseley(cmm3869 – HW03 – Gilbert...

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Unformatted text preview: moseley (cmm3869) – HW03 – Gilbert – (56380) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find a formula for the general term a n of the sequence { a n } ∞ n =1 = braceleftBig 2 , 6 , 10 , 14 , . . . bracerightBig , assuming that the pattern of the first few terms continues. 1. a n = n + 4 2. a n = 4 n − 2 correct 3. a n = n + 3 4. a n = 5 n − 3 5. a n = 3 n − 1 Explanation: By inspection, consecutive terms a n- 1 and a n in the sequence { a n } ∞ n =1 = braceleftBig 2 , 6 , 10 , 14 , . . . bracerightBig have the property that a n − a n- 1 = d = 4 . Thus a n = a n- 1 + d = a n- 2 + 2 d = . . . = a 1 + ( n − 1) d = 2 + 4( n − 1) . Consequently, a n = 4 n − 2 . keywords: 002 10.0 points Find a formula for the general term a n of the sequence { a n } ∞ n =1 = braceleftBig 1 , − 4 5 , 16 25 , − 64 125 , . . . bracerightBig , assuming that the pattern of the first few terms continues. 1. a n = parenleftBig − 5 6 parenrightBig n- 1 2. a n = − parenleftBig 5 4 parenrightBig n 3. a n = − parenleftBig 5 6 parenrightBig n 4. a n = parenleftBig − 5 4 parenrightBig n- 1 5. a n = parenleftBig − 4 5 parenrightBig n- 1 correct 6. a n = − parenleftBig 4 5 parenrightBig n Explanation: By inspection, consecutive terms a n- 1 and a n in the sequence { a n } ∞ n =1 = braceleftBig 1 , − 4 5 , 16 25 , − 64 125 , . . . bracerightBig have the property that a n = ra n- 1 = parenleftBig − 4 5 parenrightBig a n- 1 . Thus a n = ra n- 1 = r 2 a n- 2 = . . . = r n- 1 a 1 = parenleftBig − 4 5 parenrightBig n- 1 a 1 . Consequently, a n = parenleftBig − 4 5 parenrightBig n- 1 since a 1 = 1. keywords: sequence, common ratio 003 10.0 points moseley (cmm3869) – HW03 – Gilbert – (56380) 2 Compute the value of lim n →∞ 4 a n b n 6 a n − 4 b n when lim n →∞ a n = 6 , lim n →∞ b n = − 2 . 1. limit doesn’t exist 2. limit = − 12 11 correct 3. limit = − 25 22 4. limit = 25 22 5. limit = 12 11 Explanation: By properties of limits lim n → 2 4 a n b n = 4 lim n →∞ a n lim n →∞ b n = − 48 while lim n →∞ (6 a n − 4 b n ) = 6 lim n →∞ a n − 4 lim n →∞ b n = 44 negationslash = 0 . Thus, by properties of limits again, lim n →∞ 4 a n b n 6 a n − 4 b n = − 12 11 . 004 10.0 points Determine if the sequence { a n } converges, and if it does, find its limit when a n = 6 n + ( − 1) n 5 n + 5 . 1. converges with limit = 1 2. sequence does not converge 3. converges with limit = 7 5 4. converges with limit = 3 5 5. converges with limit = 6 5 correct Explanation: After division by n we see that a n = 6 + ( − 1) n n 5 + 5 n ....
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Gilbert_Hmwk03sol - moseley(cmm3869 – HW03 – Gilbert...

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