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# [分享] 数列和

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 先从简单想起。 1+2+3+4+5+6+7+8+9+10+11+12+13 =1×2/2+2×2/2+3×2/2+4×2/2+5×2/2 =(1×2/2+2×2/2)+3×2/2+4×2/2+5×2/2 =(2×3)/2+3×4/2+4×2/2+5×2/2 =(2×3/2+3×2/2)+4×2/2+5×2/2 =(3×4)/2+4×2/2+5×2/2 =(3×4/2+4×2/2)+5×2/2 =(4×5)/2+5×2/2+6×2/2 =(4×5/2+5×2/2)+6×2/2 =(5×6)/2+6×2/2+7×2/2 =(5×6/2+6×2/2)+7×2/2 =(6×7)/2+7×2/2+8×2/2 =(6×7/2+7×2/2)+8×2/2 =(7×8)/2+8×2/2+9×2/2 1×2+2×3+3×4+4×5+5×6+6×7+7×8+8×9 =1×2×3/3+2×3×3/3+3×4×3/3+4×5×3/3 =(1×2×3/3+2×3×3/3)+3×4×3/3+4×5×3/3 =(2×3×4)/3+3×4×3/3+4×5×3/3+5×6×3/3 =(2×3×4/3+3×4×3/3)+4×5×3/3+5×6×3/3 =(3×4×5)/3+4×5×3/3+5×6×3/3 =(3×4×5/3+4×5×3/3)+5×6×3/3 =(4×5×6)/3+5×6×3/3+6×7×3/3 =(4×5×6/3+5×6×3/3)+6×7×3/3 =(5×6×7)/3+6×7×3/3+7×8×3/3 =(5×6×7/3+6×7×3/3)+7×8×3/3 =(6×7×8)/3+7×8×3/3+8×9×3/3 1×2×3+2×3×4+3×4×5+4×5×6+5×6×7+6×7×8+7×8×9 =1×2×3×4/4+2×3×4×4/4+3×4×5×4/4+4×5×6×4/4 =(1×2×3×4/4+2×3×4×4/4)+3×4×5×4/4+4×5×6×4/4 =(2×3×4×5)/4+3×4×5×4/4+4×5×6×4/4+5×6×7×4/4 =(2×3×4×5/4+3×4×5×4/4)+4×5×6×4/4+5×6×7×4/4 =(3×4×5×6)/4+4×5×6×4/4+5×6×7×4/4 =(3×4×5×6/4+4×5×6×4/4)+5×6×7×4/4 =(4×5×6×7)/4+5×6×7×4/4+6×7×8×4/4 =(4×5×6×7/4+5×6×7×4/4)+6×7×8×4/4 =(5×6×7×8)/4+6×7×8×4/4+7×8×9×4/4 1×2×3×4+2×3×4×5+3×4×5×6+4×5×6×7+5×6×7×8+6×7×8×9 =1×2×3×4×5/5+2×3×4×5×5/5+3×4×5×6×5/5+4×5×6×7×5/5 =(1×2×3×4×5/5+2×3×4×5×5/5)+3×4×5×6×5/5+4×5×6×7×5/5 =(2×3×4×5×6)/5+3×4×5×6×5/5+4×5×6×7×5/5+5×6×7×8×5/5 =(2×3×4×5×6/5+3×4×5×6×5/5)+4×5×6×7×5/5+5×6×7×8×5/5 =(3×4×5×6×7)/5+4×5×6×7×5/5+5×6×7×8×5/5 =(3×4×5×6×7/5+4×5×6×7×5/5)+5×6×7×8×5/5 =(4×5×6×7×8)/5+5×6×7×8×5/5+6×7×8×9×5/5

楼主| 发表于 2020-2-9 12:39:04 | 显示全部楼层

楼主| 发表于 2020-2-9 13:03:48 | 显示全部楼层

 我只会玩这些简单的。 1^2=1+0×1 2^2=2+1×2 3^2=3+2×3 4^2=4+3×4 5^2=5+4×5 6^2=6+5×6 7^2=7+6×7 8^2=8+7×8 9^2=9+8×9 ............ $$\D\sum_{i=1}^n n^2=\frac{n(n+1)}{2}+\frac{(n-1)n(n+1)}{3}=\frac{n(n+1)(2n+1)}{6}$$

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GMT+8, 2020-4-8 09:15 , Processed in 0.058328 second(s), 19 queries .