Answer: 0.
- A 2ⁿ
- B 0
- C 1
- D -1
Correct answer: B. 0
Explanation: Substitute x=−1 into the binomial theorem (1+x)n=Σ(r=0 to n)nCr·xʳ: LHS=(1−1)n=0 (n≥1). RHS=C0−C1+C2−…+(−1)n Cn. Therefore the alternating sum =0 for all 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