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Homework #8

# Homework #8 - r be written as n 4 = 3 k 1 for some integer...

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MATH 210-02 (Fall 2007) Dr. Kwong Homework 8 (60 Points) Due Monday, 11/5/2007 . No late homework will be accepted. Instructions : Be sure to explain how you obtain your answers, and write up your solutions neatly and clearly, in a manner that could be understood by your fellow classmates. Sloppy work will be returned ungraded. Be sure to start with a draft before copying the final version to the sheets to be handed in. 1. [4 points] Let n be an integer expressible as n = 5 q + 4 for some integer q . Show that n 2 can be written as n 2 = 5 k + 1 for some integer k . 2. [6 points] Let m and n be two odd integers. Use the definition of odd and even numbers to prove that mn ( m + n ) is even. 3. [8 points] If n is an integer that is not divisible by 3, then, when n is divided by 3, what could be its remainders? In other words, if we write n = 3 q + r , where r denotes the remainder, what are the possible values of r ? Based on the possibilities, show that n 4 can always (regardless of the possible values of
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Unformatted text preview: r ) be written as n 4 = 3 k + 1 for some integer k . 4. [8 points] Prove that √ 7 is irrational. Hint : Use a proof by contradiction. Also recall that an integer m is divisible by 7 means that we can write m = 7 q for some integer q . 5. [8 points] Use induction to prove that 1 3 + 2 3 + 3 3 + ··· + n 3 = n 2 ( n + 1) 2 4 for all integers n ≥ 1. Remark : Since n 2 ( n +1) 2 4 = h n ( n +1) 2 i 2 , and we have learned that ∑ n i =1 i = n ( n +1) 2 , Problem 5 leads to a fascinating result n X i =1 i 3 = ˆ n X i =1 i ! 2 . 6. [8 points] Use induction to prove that n X i =1 (2 i + 5) = n 2 + 6 n for all integers n ≥ 1. 7. [8 points] Use induction to prove that 3 + 3 · 5 + 3 · 5 2 + ··· + 3 · 5 n-1 = 3(5 n-1) 4 for all integers n ≥ 1. Suggestion : Instead of using sigma notation, it may be easier to write out the full equation....
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