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 n1 = 3(5 n1) 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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 Fall '08
 Kwong
 Math, Division, Remainder, Natural number, 3k, integer k.

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