1/1x2x3+1/2x3x4+1/3x4x5+.+1/8x9x10 求简算

1×2×3+2×3×4+3×4×5+……+8×9×10
=(1/4)(1×2×3×4)+(1/4)(2×3×4×5-1×2×3×4)+(1/4)(3×4×5×6-2×3×4×5)+……(1/4)(8×9×10×11-7×8×9×10)
=(1/4)(1×2×3×4+2×3×4×5-1×2×3×4+3×4×5×6-2×3×4×5+……+8×9×10×11-7×8×9×10)
=(1/4)(8×9×10×11)
=2×9×10×11
=1980

1/2x3X4=（1/2X3-1/3X4）X1/21/3X4x5=（1/3X4-1/4X5）X1/2.1/8X9X10=（1/8X9-1/9X10）X1/2上述各式相加得：1/2x3X4十1/3X4x5十1/4X5X6……十1/8X9X10=（1/2X3-1/3X4+1/3X4-1/4X5+.+1/8X9-1/9X10）X1/2=（1/2X3-1/9X1...

8x9x10x11x12x13x14=63x64x65x66
因为8~14之间有11和13两个质数,且结果里面的数字越大则数字个数越少,并且不能带有质数,且结果只必有两个数字是以11和13为因数,所以开始一组一组实验,22~26中23为质数,26~33中31为质数,33~39中有34=2x17（17为质数）37也为质数,39~44中41为质数,44~52中51为质数,52~55中53为质数,55~65显然乘积过大,刚好65和66连在一起且存在63x64x65x66符合题意,故这些连续自然数中最小的数是63

1/1x2x3+1/2x3x4+1/3x4x5+1/4x5x6+...+1/8x9x10
=1-1/2-1/2(1-1/3)
+1/2-1/3-1/2(1/2-1/4)
+1/3-1/4-1/2(1/3-1/5)
+1/4-1/5-1/2(1/4-1/6)
+.
+1/8-1/9-1/2(1/8-1/10)
=1-1/9-1/2(1+1/2-1/9-1/10)
=173/360

2/1x2x3+2/2x3x4+2/3x4x5+.2/8x9x10=
=2{1/2[1/1*2-1/2*3]+1/2[1/2*3-1/3*4]+1/2[1/3*4-1/4*5]+...+1/2[1/8*9-1/9*10] }
=44/90
=22/45