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Discrete Mathematics with Graph Theory (3rd Edition) 137

# Discrete Mathematics with Graph Theory (3rd Edition) 137 -...

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Section 5.2 135 30. al = 1, a2 = al = 1, a3 = al + a2 = 2, a4 = al + a2 + a3 = 4, a5 = al + a2 + a3 + a4 = 8, a6 = al + a2 + a3 + a4 + a5 = 16. We guess an = 2 n - 2 for n ::::: 2 and prove this result by mathematical induction (strong form). For n = 2, a2 = 1 while 2 2 - 2 = 2 0 = 1 also, so the formula is correct. Now assume that k > 2 and the formula is true for all n, 2 S n < k. Then k-l k-l ak = L ai = al + L ai i=1 i=2 k-l 1 + L 2 i - 2 using al = 1 and the induction hypothesis i=2 1(1 - 2 k - 2 ) = 1 + 1 + 2 + 4 + . .. + 2 k - 3 = 1 + = 1 + (2 k - 2 - 1) = 2 k - 2 1-2 which is the desired formula with n = k. (Note that we used formula (8) to sum k - 2 terms of the geometric sequence with a = 1, r = 2 at the second last step.) We conclude that the formula is true for all n ::::: 2, by the Principle of Mathematical Induction. 7 (1-8 18 ) 7 (2 54 _1) 31. (a) The sum is 1024 1 _ 8 = 2 10 7 = 244 - T 10 ~ 244. (b) [BB]
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