Question

13．用二项式定理证明：（n+1）ⁿ-1能被n²整除．

Answer 1

分析 把（n+1）ⁿ-1按照二项式定理展开，再提取公因式，即可证明（n+1）ⁿ-1能被n²整除．

解答 证明：（n+1）ⁿ-1=${C}_{n}^{0}$•nⁿ+${C}_{n}^{1}$•n^n-1+${C}_{n}^{2}$•n^n-2+…+${C}_{n}^{n-1}$•n+${C}_{n}^{n}$-1
=${C}_{n}^{0}$•nⁿ+${C}_{n}^{1}$•n^n-1+${C}_{n}^{2}$•n^n-2+…+${C}_{n}^{n-1}$•n=n²•（${C}_{n}^{0}$•n^n-2+${C}_{n}^{1}$•n^n-3+${C}_{n}^{2}$•n^n-4+…+1），
由于${C}_{n}^{0}$•n^n-2+${C}_{n}^{1}$•n^n-3+${C}_{n}^{2}$•n^n-4+…+1为正整数，
故（n+1）ⁿ-1能被n²整除．

点评 本题主要考查二项式定理的应用，证明整除性问题，属于基础题．