Question

完全平方公式例题

及其步骤和答案
（1-1/22）（1-1/32）（1-1/42）···（1-1/992）（1-1/1002）
（括号里后面的2是平方~打不出来）

Answer 1

(1-1/2²)(1-1/3²)(1-1/4²)......(1-1/99²)(1-1/100²)
=(2²-1)/2²*(3²-1)/3²*(4²-1)/4²*......(99²-1)/99²*(100²-1)/100²
=(1*3/2²)*(2*4/3²)*(3*5/4²)*......(98*100/99²)*(99*101/100²)
=(1*2*3*4*......98*99)*(3*4*5*6*......100*101)/(2²*3²*4²*......99²*100²)
=(2*3*4*......98*99*100/100)*(2*3*4*5*6*......100*101/2)/(2*3*4*......99*100)²
=(2*3*4*......99*100)²*101/(100*2)/(2*3*4*......99*100)²
=101/200

或
(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/99²)(1-1/100²)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)....(1-1/99)(1+199)(1-1/100)(1+1/100)
=1/2*3/2*2/3*4/3*3/4*5/4*.....98/99*100/99*99/100*101/100
=1/2*101/100
=101/200