Answer: (2ⁿ⁺¹-1)/(n+1).
- A (2ⁿ+1)/(n+1)
- B 2ⁿ/(n+1)
- C (2ⁿ⁺¹-1)/(n+1)
- D 2ⁿ⁻¹/(n+1)
Correct answer: C. (2ⁿ⁺¹-1)/(n+1)
Explanation: Integrate (1+x)n from 0 to 1: [(1+x)n+1/(n+1)] from 0 to 1 = (2n+1-1)/(n+1). LHS integral = C0 + C1/2 + C2/3 + ... + Cn/(n+1). So answer = (2n+1-1)/(n+1).
Row n of Pascal's triangle gives the coefficients nC0, nC1, ..., nCn for the expansion of (a+b)ⁿ; each number is the sum of the two numbers diagonally above it, a direct visual proof of the identity nCr = (n-1)C(r-1) + (n-1)Cr.
Concept context
Expansion of (a+b) n , general term, middle term, binomial coefficients, and greatest term