Induction Proofs

# Induction Proofs - Math 100 Induction and Sums 1 Some Sums...

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Math 100 Induction and Sums Martin H. Weissman 1. Some Sums Suppose that a 0 ,a 1 ,a 2 ,... is a sequence of real numbers. Then, one may create a new sequence, by summation: for all natural numbers n , deﬁne: s n = n X i =0 a i . One could deﬁne the sequence s n recursively as following: The ﬁrst term is deﬁned by s 0 = a 0 . For all n N , if n > 0, then s n = a n + s n - 1 . First, we will experiment with summation. Given the following sequences, determine the sequence of sums: (1) 1 , 2 , 4 , 8 , 16 , 32 ,... (2) 1 , 3 , 5 , 7 , 9 , 11 ,... (3) 0 , 1 , 2 , 3 , 4 , 5 , 6 ,... (4) 1 / 2 , 1 / 6 , 1 / 12 , 1 / 20 ,... (5) 1 , 1 / 2 , 1 / 4 , 1 / 8 , 1 / 16 ,... Can you guess formulas for the sequences of sums? Today, we’ll spend some time proving these formulas. After guesses are made, we’ll prove the guesses, using the following proof as a template. Theorem 1. Let a 0 ,a 1 ,a 2 ,... be the sequence, which satisﬁes a n = 2 n for all natural numbers n . Let s n = n i =0 a

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## This note was uploaded on 04/17/2008 for the course MATH 100 taught by Professor Weissman during the Fall '08 term at UCSC.

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Induction Proofs - Math 100 Induction and Sums 1 Some Sums...

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