Question

一道用乘法公式计算的数学题

分母上的指数是2

Answer 1

因为公式：1-1/n^2=[（n-1）/n]×[（n+1）/n]
然后一一带入即可，即(1-1/2²)(1-1/3²)(1-1/4²).......(1-1/100²)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4).......(1-1/100)(1+1/100)
=1/2×3/2×2/3×4/3×3/4×5/4×......×100/99×99/100×101/100
=1/2×101/100
=101/200

Answer 2

1-1/2²=(1-1/2)(1+1/2)=(1/2)×(3/2)
1-1/3²=(1-1/3)(1+1/3)=(2/3)×(4/3)
…………
1-1/99²=(1-1/99)(1+1/99)=(98/99)×(100/99)
1-1/100²=(1-1/100)(1+1/100)=(99/100)×(101/100)
原式=(1/2)×(3/2))×(2/3)×(4/3)×……×(98/99)×(100/99)×(99/100)×(101/100)=101/200

Answer 3

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