Question

find the following expression 1/(2^2 -1 ) + 1/(4^2 - 1)+.....+ 1/(20^2 - 1)

Answer 1

Note that 2^2 - 1 is (2-1)(2+1), 4^2 - 1 is (4-1)(4-1) ... so on

and

1 = 1/2 {(2+1) - (2-1)} = 1/2 {(4+1) - (4-1)} = ... so on.

Hence,

=1/2[{(2+1) - (2-1)}/(2^2 - 1) + {(4+1) - (4-1)}/(4^2 - 1) + ... + {(20+1) - (20-1)}/(20^2 - 1)]

=1/2[1/(2-1) - 1/(2+1) + 1/(4-1) - 1/(4+1) + ... + 1/(20-1) - 1/(20+1)]

=1/2[1 - 1/21]

=10/21

Answer 2

{1/(2^2-1}+{1/(4^2-1}+{1/(6^2-1}+.......+{1/(20^2-1}

= 1/3 +1/15 +.....+1/399

= (1/1-2/3)+(2/3-3/5) +...+(10/19-11/21)

= 1-11/21

=10/21