quiz5_key

quiz5_key - COT 3100 Summer 2001 Quiz #4 (Solutions)...

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COT 3100 Summer 2001 Quiz #4 (Solutions) 07/12/2001 Total 25 pts 1. (7pts) Use induction to show that 5 n - 1 is divisible by 4 for n =1, 2, … Basis: n =1, LHS=5 1 - 1=4 is divisible by 4. IH. Assume that for n = k , k is some integer k 1, the proposition is true, i. e. 5 k - 1 =4 s , s is any integer. IS . We need to prove that 5 k +1 - 1 is divisible by 4, i.e. 5 k +1 - 1 = 4 p , p is any integer . 5 k +1 - 1 = 5 (5 k - 1)+4 =5 4 s +4 (by IH) = 4 (5 s +1) = 4 p , where p = 4 s + 1, an integer. 2. (8pts) Prove that 2 n > n 2 whenever n is an integer greater than 4. Basis . n = 5, 2 5 =32> 5 2 =25. IH . Assume for n = k , where k is some integer k >4, the inequality holds, i.e. 2 k > k 2 . IS . We need to prove that the inequality holds for n = k +1, i.e. 2 k +1 >( k +1) 2 . 2 k +1 = 2 2 k > 2 k 2 (by IH) = k 2 + k 2 > k 2 +2 k +1 =( k +1) 2 , because k 2 >2 k +1 for k >4. The reason why it is true can be implied from the fact that the quadratic polynomial
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