Let n be a positive integer. Let A = ∑ k = 0 n - 1 k × C k n 1 2 k + 3 4 k + 7 8 k + 15 16 k + 31 32 k . If 63 A = 1 - 1 2 30 , then n is equal to ______ .

Question

Let n be a positive integer. Let A = ∑ k = 0 n - 1 k × C k n 1 2 k + 3 4 k + 7 8 k + 15 16 k + 31 32 k . If 63 A = 1 - 1 2 30 , then n is equal to ______ .,

MARKS App · Accepted Answer

A = ∑ k = 0 n C k n - 1 2 k + - 3 4 k + - 7 8 k + - 15 16 k + - 31 32 k A = 1 - 1 2 n + 1 - 3 4 n + 1 - 7 8 n + 1 - 15 16 n + 1 - 31 32 n A = 1 2 n + 1 4 n + 1 8 n + 1 16 n + 1 32 n A = 1 2 n 1 - 1 2 n 5 1 - 1 2 n ⇒ A = 1 - 1 2 5 n 2 n - 1 2 n - 1 A = 1 - 1 2 5 n , Given, 63 A = 1 - 1 2 30 Clearly 5 n = 30 n = 6

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