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Let n be a positive integer. Let A = Σ from k=0 to n [(-1)^k nCk { (1/2)^k + (3/4)^k + (7/8)^k + (15/16)^k + (31/31)^k } ] If 63A = 1 − 1/2^30, then n is equal to _____.

1. 4
2. 5
3. 6
4. 7

Correct answer: 5

Solution

The expression for A involves a sum that can be simplified using the binomial theorem and properties of geometric series. The condition given, 63A = 1 − 1/2^30, leads to a specific value for n, which through calculation shows that n must equal 5 to satisfy the equation.

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