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求证1×2+2×3+3×4+…+n(n+1)=
1
3
n(n+1)(n+2)
人气:145 ℃ 时间:2019-11-06 19:03:10
解答
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,
1×2+2×3+3×4+…+k(k+1)=
1
3
k(k+1)(k+2)

则当n=k+1时,
左边=
1
3
k(k+1)(k+2)+(k+1)(k+2)
=(k+1)(k+2)(
1
3
k+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
3
n(n+1)(n+2)
对任意正整数都成立.
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