hw3solns - Math 55 Discrete Mathematics UC Berkeley Fall 2011 Homework 3 due Wedneday February 8 February 8 2012 2.4.26 For each of these lists of

hw3solns - Math 55 Discrete Mathematics UC Berkeley Fall...

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Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 3, due Wedneday, February 8 February 8, 2012 2.4. 26 For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. We will refer to the terms as a 1 , a 2 , a 3 , ... a) 3 , 6 , 11 , 18 , 27 , 38 , 51 , 66 , 83 , 102 , ... The pattern is a n +1 = a n + 2 n + 1. The next three terms are 123 , 146 , 171 . b) 7 , 11 , 15 , 19 , 23 , 27 , 31 , 35 , 39 , 43 , ... The pattern is a n +1 = a n + 4 (which gives the formula a n = 4 n + 3). The next three terms are 47 , 51 , 55. c) 1 , 10 , 11 , 100 , 101 , 110 , 111 , 1000 , 1001 , 1010 , 1011 , ... The pattern is running through all binary strings in order. (Equiv- alently: it is the sequence of natural numbers, written in binary.) The next three terms are 1100, 1101, 1110. d) 1 , 2 , 2 , 2 , 3 , 3 , 3 , 3 , 3 , 5 , 5 , 5 , 5 , 5 , 5 , 5 , ... This sequence is made up of Fibonacci numbers, starting with one 1, then three 2’s then five 3’s, and so on, increasing the number of copies by two each time. The next three terms are 8 , 8 , 8. e) 0 , 2 , 8 , 26 , 80 , 242 , 728 , 2186 , 6560 , 19682 , ... The pattern is a n +1 = 3 a n + 2. The next three terms are 59048, 177146, and 531440. f) 1 , 3 , 15 , 105 , 945 , 10395 , 135135 , 2027025 , ... The pattern is a n +1 = (2 n + 1) · a n (which gives the formula a n = (2 n - 1) · (2 n - 3) · ... · 5 · 3 · 1). The next three terms are 34459425 , 654729075 , 13749310575. 1
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g) 1 , 0 , 0 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 1 , ... The pattern is one 1, two 0’s, three 1’s, four 0’s, and so on. The next three terms are 0 , 0 , 0. h) 2 , 4 , 16 , 256 , 65536 , 4294967296 , ... The pattern is a n +1 = a 2 n . The next three terms are 18446744073709551616 340282366920938463463374607431768211456 115792089237316195423570985008687907853269984665640564039457584007913129639936 2.4. 32 Compute the value of each of these sums. a) 8 X j =0 (1 + ( - 1) j ) =1 + 0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 = 5 b) 8 X j =0 (3 j - 2 j ) = 8 X j =0 3 j - 8 X j =0 2 j = 3 9 - 1 3 - 1 - 2 9 - 1 2 - 1 =9330 . c) 8 X j =0 (2 · 3 j + 3 · 2 j ) =2 · 8 X j =0 3 j + 3 · 8 X j =0 2 j =2 · 3 9 - 1 3 - 1 + 3 · 2 9 - 1 2 - 1 =21215 .
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