Exam_1_Solution

Exam_1_Solution - Ma1a First Test Solutions Fall 2005...

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Unformatted text preview: Ma1a First Test Solutions Fall 2005 Problem 1. (Brian) Prove that for n = 2 , 3 , . . . , 1- 1 4 1- 1 9 1- 1 n 2 = n + 1 2 n . Solution. Let A = n Z + : n 2 & 1- 1 4 1- 1 9 1- 1 n 2 = n + 1 2 n . We will show that A = { n Z + : n 2 } by induction. First, note that ( 1- 1 4 ) = 3 4 = 2+1 2 2 , so 2 A . Now suppose n A . Then we have 1- 1 4 1- 1 9 1- 1 n 2 = n + 1 2 n , 1- 1 4 1- 1 9 1- 1 n 2 1- 1 ( n + 1) 2 = n + 1 2 n 1- 1 ( n + 1) 2 , 1- 1 4 1- 1 9 1- 1 n 2 1- 1 ( n + 1) 2 = n + 1 2 n n 2 + 2 n ( n + 1) 2 , 1- 1 4 1- 1 9 1- 1 n 2 1- 1 ( n + 1) 2 = n ( n + 1)( n + 2) 2 n ( n + 1) 2 , 1- 1 4 1- 1 9 1- 1 n 2 1- 1 ( n + 1) 2 = ( n + 1) + 1 2( n + 1) . so n + 1 A . So we have 2 A and if n A then n + 1 A . So by the principle of mathematical induction, A = { n Z + : n 2 } , and the formula is true for all n = 2 , 3 , . . . . Problem 2. (Dr. Simon) (a) Give the N definition of x n x ....
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Exam_1_Solution - Ma1a First Test Solutions Fall 2005...

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