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Question 130
What is the value of the following expression?
$$(1/(2^2-1))+(1/(4^2-1))+(1/(6^2-1))+...+(1/(20^2-1)$$
Solution
$$(1/(2^2-1))+(1/(4^2-1))+(1/(6^2-1))+...+(1/(20^2-1)$$ = 1/[(2+1)*(2-1)] + 1/[(4+1)*(4-1)] + ... + 1/[(20+1)*(20-1)]
= 1/(1*3) + 1/(3*5) + 1/(5*7) + ... + 1/(19*21)
=1/2 * ( 1/1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + ... +1/19 - 1/21)
=1/2 * (1 - 1/21) = 10/21
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