Question

求a1+a2+a3+......+a100的值

a1=1/1X3=1/2X(1-1/3)
a2=1/3X5=1/2X(1/3-1/5)
a3=1/5x7=1/2x(1/5-1/7)

Answer 1

a1+a2+a3+......+a100

=

=(1/2 )*(200/201)

=100/201

Answer 2

a1+a2+a3+……+a100
=1/2×(1-1/3)+1/2×(1/3-1/5)+1/2×(1/5-1/7)+……+1/2×(199-1/201)
=1/2×(1-1/3+1/3-1/5+1/5-1/7+……+1/199-1/201)
=1/2×(1-1/201)
=1/2×200/201
=100/201

Answer 3

an=1/[(2n-1)(2n+1)]=(1/2)[1/(2n-1)-1/(2n+1)]
a1+a2+...+a100
=(1/2)[1/1-1/3+1/3-1/5+...+1/(2×100-1)-1/(2×100+1)]
=(1/2)(1- 1/201)
=100/201