Math 347 Worksheet : Induction Proofs , I — Solutions
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چکیده
1. Induction proofs, type I: Sum/product formulas: The most common, and the easiest, application of induction is to prove formulas for sums or products of n terms. All of these proofs follow the same pattern. (a) ∑n i=1 i(i+ 1) = n(n+1)(n+2) 3 (b) ∑n i=0 2 i = 2n+1 − 1 (sum of powers of 2) (c) ∑n i=0 r i = 1−r n+1 1−r (r 6= 1) (sum of finite geometric series) (d) ∑n i=0 i!i = (n+ 1)!− 1. Solution: All proofs follow the pattern illustrated by the sample proof (of the formula ∑n i=1 i = n(n+1)/2). We will carry out the details for (a) and (d). The other formulas can be proved similarly. (Note that (b) is a special case of (c).) Proof of (a): We seek to show that, for all n ∈ N, (∗) n ∑
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Math 347 Worksheet : Induction Proofs , II — Solutions A
(For convenience, we define F0 = 0; with this definition, the recurrence relation Fn = Fn−1 + Fn−2 holds for all n ≥ 2 and the above matrix is well defined for all n ≥ 1.) Base case: When n = 1, the four entries of the matrix on the right are F2 = 1, F1 = 1, F1 = 1, and F0 = 0, so (∗) holds in this case. Induction step: Let k ∈ N be given and suppose (∗) holds for n = k. Then ( 1 1 1 0 )k+1 = (...
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