Sum of `1/(sqrt(2)+sqrt(5))+1/(sqrt(5)+sqrt(8))+1/(sqrt(8)+sqrt(11))+1/(sqrt(11)+sqrt(14))+..to n` terms= (A) `n/(sqrt(3n+2)-sqrt(2))` (B) `1/3 (sqrt(2)-sqrt(3n+2)` (C) `n/(sqrt(3n+2)+sqrt(2))` (D) none of these

Options

A. `((sqrt(3n +2))-sqrt(2))/(3)`
B. `(n)/(sqrt(2+3n)+sqrt(2))`
C. less than n
D. less than `sqrt(n/3)`

Answer

A, B, C

Explanation

`1/(sqrt2+sqrt5)+1/(sqrt5+sqrt8)+1/(sqrt8+sqrt(11))+…` n terms
`=(sqrt5-sqrt2)/3+(sqrt8-sqrt5)/3+….+(sqrt(5+(n-1)3)-sqrt(2+(n-1)3))/3`
`=(sqrt(3n+2)-sqrt2)/3`
`=(3n+2-2)/(3(sqrt(3n+2)+sqrt2))`
`=n/(sqrt(3n+2)+sqrt2)`
`=n/(sqrt(2+3n)+sqrt2)ltn/(sqrt(3n))ltn`