9 Prove that 1 2 2 2 2 n 1 2 n 1 for all n N Let S be the set of positive

9 prove that 1 2 2 2 2 n 1 2 n 1 for all n n let s be

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9. Prove that 1 + 2 + 2 2 + · · · + 2 n - 1 = 2 n - 1 for all n N . Let S be the set of positive integers for which the equation holds. Step 1. 1 S : 1 = 2 1 - 1 = 1. Therefore, 1 S . Step 2. Assume that the positive integer k S . That is, assume 1 + 2 + 2 2 + · · · + 2 k - 1 = 2 k - 1 . Step 3. Prove that k + 1 S . 1 + 2 + 2 2 + · · · + 2 k - 1 + 2 k ? = 2 k +1 - 1 2 k - 1 + 2 k ? = 2 k +1 - 1 2 · 2 k - 1 ? = 2 k +1 - 1 2 k +1 - 1 = 2 k +1 - 1 Therefore, k + 1 S and S = N . 19. Prove that parenleftbigg 1 - 1 2 2 parenrightbigg · parenleftbigg 1 - 1 3 2 parenrightbigg · · · parenleftbigg 1 - 1 n 2 parenrightbigg = n + 1 2 n for all n 2. Let S be the set of positive integers for which the equation holds. Step 1. 2 S : ( 1 - 1 2 2 ) = 3 4 2+1 2 · 2 . Therefore, 2 S . Step 2. Assume that the positive integer k S . That is, assume parenleftbigg 1 - 1 2 2 parenrightbigg · parenleftbigg 1 - 1 3 2 parenrightbigg · · · parenleftbigg 1 - 1 k 2 parenrightbigg = k + 1 2 k . Step 3. Prove that k + 1 S . parenleftbigg 1 - 1 2 2 parenrightbigg · parenleftbigg 1 - 1 3 2 parenrightbigg · · · parenleftbigg 1 - 1 k 2 parenrightbigg · parenleftbigg 1 - 1 ( k + 1) 2 parenrightbigg ? = k + 2 2( k + 1) k + 1 2 k · parenleftbigg 1 - 1 ( k + 1) 2 parenrightbigg ?
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  • Fall '08
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  • Integers, Natural number, Prime number

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