math 311w homework2

# math 311w homework2 - a + b ) n as ( n i ) . ( a + b ) n =...

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Homework 2. Due Monday, January 31, 2010 Jaunary 25, 2009 Give rigorous solutions of all problems. Problem 1. Prove, using mathematical induction that for n 2: 1 + 1 2 + 1 3 + ··· + 1 n > n Problem 2. We denote as F n the n -th Fibonacci’s number, where F 0 = 0 ,F 1 = 1 ,F 2 = 1 ,F 3 = 2 ,F 4 = 3 ,... . We want to prove an explicit formula for calculating F n . Let λ 1 = 1 + 5 2 , λ 2 = 1 - 5 2 Number λ 1 is called ”golden ratio”, λ 2 =-1/ λ 1 a) Prove that λ 1 and λ 2 are roots of the quadratic equation: x 2 = x + 1 b) Prove that λ n 1 = λ n - 1 1 + λ n - 2 1 and λ n 2 = λ n - 1 2 + λ n - 2 2 for all n 2. c) Prove that F n = λ n 1 - λ n 2 5 for all n 0 1

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Remark: In the formula above both numbers λ 1 and λ 2 are irrational, but the result is always an integer number. Problem 3. Find the expansion for: a) ( x + y ) 8 = b) ( x - y ) 7 = We denote the coeﬃcients in the expansion of (
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Unformatted text preview: a + b ) n as ( n i ) . ( a + b ) n = ± n ² a n + ± n 1 ² a n-1 b + ± n 2 ² a n-2 b 2 + ··· + ± n n-1 ² ab n-1 + ± n n ² b n or ( a + b ) n = n X i =0 ± n i ² a n-i b i From the proof in the class we know that coeﬃcients ( n i ) satisfy ± n + 1 ² = ± n ² , ± n + 1 n + 1 ² = ± n n ² , and for 1 ≤ i ≤ n ± n + 1 i ² = ± n i-1 ² + ± n i ² Problem 4. Prove that n X i =0 ± n i ² = 2 n n X i =0 (-1) i ± n i ² = 0 Problem 5. Prove, using mathematical induction that ± n 1 ² + 2 ± n 2 ² + 3 ± n 3 ² + ··· + n ± n n ² = n 2 n-1 2...
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## This note was uploaded on 10/13/2010 for the course MATH 311W taught by Professor Mullen during the Spring '08 term at Penn State.

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math 311w homework2 - a + b ) n as ( n i ) . ( a + b ) n =...

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