计算1×2²+2×3²+3×4²+4×5²+······+18×19²+19×20²
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/17 17:54:19
![计算1×2²+2×3²+3×4²+4×5²+······+18×19²+19×20²](/uploads/image/z/201406-22-6.jpg?t=%E8%AE%A1%E7%AE%971%C3%972%26%23178%3B%2B2%C3%973%26%23178%3B%2B3%C3%974%26%23178%3B%2B4%C3%975%26%23178%3B%2B%C2%B7%C2%B7%C2%B7%C2%B7%C2%B7%C2%B7%2B18%C3%9719%26%23178%3B%2B19%C3%9720%26%23178%3B)
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计算1×2²+2×3²+3×4²+4×5²+······+18×19²+19×20²
计算1×2²+2×3²+3×4²+4×5²+······+18×19²+19×20²
计算1×2²+2×3²+3×4²+4×5²+······+18×19²+19×20²
n(n+1)²=n(n+1)(n+2)-n(n+1)
原式=1×2×3+2×3×4+...+19×20×21-(1×2+2×3+...+19×20)
1×2×3=(1×2×3×4-0×1×2×3)÷4
2×3×4=(2×3×4×5-1×2×3×4)÷4
.
1×2=(1×2×3-0×1×2)÷3
2×3=(2×3×4-1×2×3)÷3
.
原式=19×20×21×22÷4-19×20×21÷3
=43890-2660
=41230