Question

13．用二項式定理證明：（n+1）ⁿ-1能被n²整除．

Answer 1

分析 把（n+1）ⁿ-1按照二項式定理展開，再提取公因式，即可證明（n+1）ⁿ-1能被n²整除．

解答 證明：（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為正整數(shù)，
故（n+1）ⁿ-1能被n²整除．

點評 本題主要考查二項式定理的應(yīng)用，證明整除性問題，屬于基礎(chǔ)題．