Discrete Mathematics with Graph Theory (3rd Edition) 208

Discrete Mathematics with Graph Theory (3rd Edition) 208 -...

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206 Solutions to Exercises Using the Binomial Theorem, (1 + v'2t = 1 + nv'2 + (~) (v'2)2 + (~) (v'2)3 + (~) ( v''2)4 +(~)(v'2)5 + (~)(v'2)6 + ... + (v'2)n from which the result follows. 1 + nv'2 + 2(~) + 2v'2(~) + 4(~) + 4v'2(~) +8(~) + 8V2(~) + . .. + (V2)n (c) Xn+l + Yn+l V2 = (1 + V2)n+l = (1 + V2)(I + V2)n = (1 + V2)(xn + YnV2) = Xn + YnV2 + V2xn + 2Yn = (xn + 2Yn) + (xn + Yn)V2. Equating coefficients of 1 and V2, we have the desired formulas. 15. (a) Using the Binomial Theorem, (v'2 - I)n = (v'2)n + + (~) (v'2)n-l( -1) + (;) (v'2)n-2( _1)2 + . .. + (~)(v'2t-k(-I)k + . .. + (-It. When n - k is even, (V2)n-k = 2 n;-k, which is an integer. When n - k is odd, (V2)n-k = n-k-l V2( V2)n-k-l = V2V2- 2 - which is of the form cV2 for some integer c. It follows that every term in the expansion of (V2 - I)n is either an integer or of the form cV2. Collecting similar terms gives the result. (b) Since 1 < 2 < 4, we know 1 < V2 < 2, hence 0 < V2 - 1 < 1. So limn--> 00 ( V2 - I)n = o. (c) Suppose V2 is rational. Then V2
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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