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Evaluate the series S = nC1/2 + nC3/4 + nC5/6 +... (where nCr denotes the binomial coefficient C(n,r)), and express S in closed form.

1. 2ⁿ / (n + 1)
2. (2ⁿ + 1) / (n + 1)
3. (2ⁿ + 1) / (n - 1)
4. (2ⁿ - 1) / (n + 1)

Correct answer: (2ⁿ - 1) / (n + 1)

Solution

Integrating [(1+x)ⁿ - (1-x)ⁿ] from 0 to 1 isolates only the odd-r binomial terms and equals (2^(n+1) - 2)/(n+1), so the sum S = (2ⁿ - 1)/(n+1).

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