quiz5bANS - n 3 + 2 n whenever n is a nonnegative integer....

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Math 240, Quiz 4 Name: Circle One: T 12:05 T 2:25 R 12:05 R 2:25 Instructions: Answer all questions fully, showing work where necessary. 1) Compute each of these sums: a) 8 j =0 (3 j - 2 j ) People made this harder than it needs to be. We can break this up into 8 j =0 3 j - 8 j =0 2 j . These are both geometric sums. Hence this equals 3 9 - 1 3 - 1 - 2 9 - 1 2 - 1 = 9841 - 511 = 9330. b) 2 i =0 3 j =0 (3 j + 2 i ) Just write out the terms and get (0 + 0) + (3 + 0) + (6 + 0) + (9 + 0) + (0 + 2) + (3 + 2) + (6 + 2) + (9 + 2) + (0 + 4) + (3 + 4) + (6 + 4) + (9 + 4) = 78 2) Use mathematical induction to show that 3 divides
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Unformatted text preview: n 3 + 2 n whenever n is a nonnegative integer. Proof by induction: Base step: n = 1. Then n 3 + 2 n = 3, so 3 | n 3 + 2 n . Induction step: we assume 3 | n 3 +2 n , and we need to show 3 | ( n +1) 3 +2( n +1). Now, ( n + 1) 3 + 2( n + 1) = n 3 + 3 n 2 + 3 n + 1 + 2 n + 2. We can arrange this as n 3 + 2 n + 3 n 2 + 3 n + 3. By the inductive hypothesis, 3 | n 3 + 2 n . Obviously 3 | 3 n 2 + 3 n + 3, so 3 | ( n + 1) 3 + 2( n + 1). 1...
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