Answer: 0.
- A 2ⁿ
- B 0
- C 1
- D -1
Correct answer: B. 0
Explanation: Substitute x=−1 into the binomial theorem (1+x)<sup>n</sup>=Σ(r=0 to n)nCr·xʳ: LHS=(1−1)<sup>n</sup>=0 (n≥1). RHS=C<sub>0</sub>−C<sub>1</sub>+C<sub>2</sub>−…+(−1)<sup>n</sup> 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