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hmk5 - Prove using strong induction that g(n> 4 n for all...

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COT 3100 Homework # 5 Fall 2000 Assigned: 10/31/00 Due: 11/14/00 (in class) 1) Use Euclid’s Algorithm to find the greatest common divisor(GCD) of 245 and 455. 2) Show that if 11 | (6x + 5y), then there are no integer solutions to the equation 9x + 13y = 20000 3) Prove that if 3 | 2x + 7y, then 60 | (20x – 80y). 4) Using mod rules, find the remainder when you divide 5 192 by 7. 5) Let a and b be two positive integer. If a is odd, show that 9ab + 5b 2 is even. 6) Using induction, prove that 11 | (7 2n – 4 2n ) for all integers n 0. 7) Using induction, show the following: = n k k kx 1 = (x – (n+1)x x+1 + nx n+2 )/((1 – x) 2 ). 8) Define a recurrence relation g as follows: g(0) = 2, g(1) = 3, g(n) = 3g(n-1) + 4g(n-2), for all integers n>1.
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Unformatted text preview: Prove, using strong induction, that g(n) > 4 n , for all n>1. 9) The Fibonacci numbers are defined as follows: F 1 = F 2 = 1, and F n = F n-1 + F n-2 , for all integers n > 2. Prove the following closed summation closed form using induction: ∑ = n i i F 1 2 = F n F n+1 , for integers n > 0. 10) Let g: A → A be a bijection. For n ≥ 2, define g n = g ° g ° ... ° g, where g is composed with itself n times. Prove that for n ≥ 2, that g n is a bijection from A to A as well, and show that (g n )-1 = (g-1 ) n . (Here you may assume that the composition of two bijections is also a bijection.)...
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