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If a(n) = sqrt(1+(1+(1)/(n))^(2))+sqrt(1...

If `a_(n) = sqrt(1+(1+(1)/(n))^(2))+sqrt(1+(1-(1)/(n))^(2))` then value of `sum_(n=1)^(20) (1)/(a^(n))` is

A

even

B

odd

C

prime

D

divisible by `'3'`

Text Solution

Verified by Experts

The correct Answer is:
B, C
`(1)/(a_(n)) = (1)/(sqrt(2 + (2)/(n) + (1)/(n^(2))) + sqrt(2 - (2)/(n) + (1)/(n^(2)))`
`(n)/(sqrt(2n^(2) + 2n + 1) + sqrt(2n^(2) - 2n + 1))`
`= (n)/(4n) [sqrt(2n^(2) + n + 1) - sqrt(2n^(2) - 2n + 1)]`
`:. sum_(n=1)^(20)(1)/(a_(n)) = (1)/(4)[sqrt(841) - 1] = 7`
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