math 311w homework1

# math 311w homework1 - Homework 1 Due Friday Give rigorous...

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Due Friday, January 22, 2010 Give rigorous solutions of all problems. Problem 1. Describe all integer solutions of the equation n 2 + 2 m 2 = l 2 Hint: see Handout 1. Answer: n = 2 s 2 - t 2 ,m = 2 st,l = 2 s 2 - t 2 where s,t Z . Problem 2. Prove, using mathematical induction that 1 + 3 + 5 + 7 + ··· + 2 n - 1 = n 2 Proof. We will prove this formula by induction. First, we check the base of induction: Base of induction: 1 = 1 2 Step of induction: Assume for some number n the formula is correct: 1 + 3 + 5 + 7 + ··· + 2 n - 1 = n 2 We need to prove that 1 + 3 + 5 + 7 + ··· + 2 n - 1 + (2 n + 1) = ( n + 1) 2 In order to do it we compare how the right and left sides of the formula change when we go from n to n + 1 . Left side changes by 2 n + 1 . Right side changes by ( n + 1) 2 - n 2 = n 2 + 2 n + 1 - n 2 = 2 n + 1 . Both sides of the formula change in the same way. Therefore, if the formula is true for n then it is true for n + 1 . 1

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math 311w homework1 - Homework 1 Due Friday Give rigorous...

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