Sec. 8.1 HW

Sec. 8.1 HW - Find the ±rst 40 terms of the sequence...

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Homework 8.1 Problem 6 Find a formula for the general term a n of the sequence, assuming that the pattern of the ±rst few terms continues: {- 1 4 , 2 9 , - 3 16 , 4 25 ,..., } The numerators are as follows: For a 1 it is 1, for a 2 it is 2, for a 3 it is 3, and for a 4 it is 4. This means that the numerator of a n is n . The sign is negative for the ±rst and third terms and positive for the second and fourth terms. Thus the sign of a n is ( - 1) n . As for the denominators, for a 1 it is 4 = 2 2 , for a 2 it is 9 = 3 2 , for a 3 it is 16 = 4 2 and for a 4 it is 25 = 5 2 . Therefore, for a n the denominator is ( n + 1) 2 . Therefore, we may conclude that a n = ( - 1) n n ( n + 1) 2 Problem 16 Determine whether the sequence converges or diverges. If it converges, ±nd its limit: a n = ( - 1) n n 3 n 3 +2 n 2 +1 Notice that lim n →∞ n 3 n 3 + 2 n 2 + 1 = lim n →∞ n 3 n 3 + 2 n 2 + 1 b 1 n 3 1 n 3 B = lim n →∞ 1 1 + 2 n + 1 n 3 = 1 1 + 0 + 0 = 1 This must therefore mean that the a n will ultimately oscillate back and forth between values which are alternately close to 1 and then -1. Therefore, no single value is approached by the a n and so we must conclude that lim n →∞ a n does not exist. Problem 36
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Unformatted text preview: Find the ±rst 40 terms of the sequence de±ned by a n = ± 1 2 a n if a n is an even number 3 a n + 1 if a n is an odd number with a 1 = 11 and then do this again with a 1 = 25 . Make a conjecture about this type of sequence. Note that we can use an Excel spreadsheet to compute a n +1 from a n with the following formula when a n is the entry in cell A 1:(We conjecture that the 421 cycle is always reached) “= IF ( MOD (A1 , 2) = 0 , A1 2 , 3A1 + 1)” n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a n 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 4 2 1 4 2 n 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 a n 1 4 2 1 4 2 1 4 2 1 4 2 1 4 2 1 4 2 1 4 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a n 25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 n 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 a n 8 4 2 1 4 2 1 4 2 1 4 2 1 4 2 1 4 2 1 4...
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This homework help was uploaded on 04/08/2008 for the course MATH 182 taught by Professor Keppelmann during the Spring '08 term at Nevada.

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