Question

(1-1/2^2)(1-1/3^2)……(1-1/100^2)的结果是多少？

带过程

Answer 1

(1-1/2^2)(1-1/3^2)……(1-1/100^2)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)……(1+1/100)(1-1/100)
=（1+1/2）(1+1/3)……(1+1/100)×(1-1/2)(1-1/3)……(1-1/100)
=3/2×4/3×5/4×6/5×……×101/100×1/2×2/3×3/4×…… ×99/100
约分可得
=101/2(加号的部分)×1/100（减号的部分）
=101/200

Answer 2

用平方差
原式=(1-1/2)(1+1/2)(1-1/3)(1+1/3)……(1-1/100)(1+1/100)
=(1/2)(3/2)(2/3)(4/3)……(99/100)(101/100)
中间约分
=(1/2)(101/100)
=101/200

Answer 3

(1-1/2^2)(1-1/3^2)……(1-1/100^2)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)......(1-1/100)(1+1/100)
=1/2*3/2*2/3*4/3*...*99/100*101/100
=1/2*101/100
=101/200

Answer 4

由n^2-1=(n-1)(n+1)
(1-1/2^2)*(1-1/3^2)*(1-1/4^2).......(1-1/100^2)
=(2^2-1)/2^2*(3^2-1)/3^2........(100^2-1)/100^2
=(1*3/2^2)*(2*4/3^2)*(3*5/4^2)....99*101/100^2
=1/2*101/100(中间约了很多项)
=101/200

Answer 5

用平方差公式=[（1/2)*(3/2)]*[（2/3)*(4/3)]*...*[(99/100)*(101/100)]
=101/200