求证1×2+2×3+3×4+…+n(n+1)＝13n(n+1)(n+2)

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求证1×2+2×3+3×4+…+n(n+1)＝

n(n+1)(n+2)

证明：①当n=1时，左边=2，右边=
1
3×1×2×3＝2，等式成立；
②假设当n=k时，等式成立，
即1×2+2×3+3×4+…+k(k+1)＝
1
3k(k+1)(k+2)
则当n=k+1时，
左边=
1
3k(k+1)(k+2)+(k+1)(k+2)=（k+1）（k+2）（
1
3k+1）=
1
3（k+1）（k+2）（k+3）
即n=k+1时，等式也成立．
所以1×2+2×3+3×4+…+n(n+1)＝
1
3n(n+1)(n+2)对任意正整数都成立．

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