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)
1 |
3 |
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.
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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