Question

12．求证：${C}_{n}^{n}$+${C}_{n+1}^{n}$+…+${C}_{2n-1}^{n}$+${C}_{2n}^{n}$=${C}_{2n+1}^{n+1}$．

Answer 1

分析 利用组合数公式${C}_{n}^{m+1}$+${C}_{n}^{m}$=${C}_{n+1}^{m+1}$，即可证明等式成立．

解答 证明：∵${C}_{n}^{m+1}$+${C}_{n}^{m}$=${C}_{n+1}^{m+1}$，
∴${C}_{n}^{n}$+${C}_{n+1}^{n}$+…+${C}_{2n-1}^{n}$+${C}_{2n}^{n}$=（${C}_{n+1}^{n+1}$+${C}_{n+1}^{n}$）+${C}_{n+2}^{n}$+…+${C}_{2n-1}^{n}$+${C}_{2n}^{n}$
=${C}_{n+2}^{n+1}$+${C}_{n+2}^{n}$+…+${C}_{2n-1}^{n}$+${C}_{2n}^{n}$
=${C}_{n+3}^{n+1}$+…+${C}_{2n-1}^{n}$+${C}_{2n}^{n}$
=…=${C}_{2n-1}^{n+1}$+${C}_{2n-1}^{n}$+${C}_{2n}^{n}$
=${C}_{2n}^{n+1}$+${C}_{2n}^{n}$
=${C}_{2n+1}^{n+1}$．

点评 本题考查了组合数公式的应用问题，也考查了逻辑推理与证明的应用问题，是基础题目．