`S_(n)` be the sum of n terms of the series `(8)/(5)+(16)/(65)+(24)/(325)+"......"`
The seveth term of the series is (a)`(56)/(2505)` (b)`(56)/(6505)` (c)`(56)/(5185)` (d)`(107)/(9605)`

Options

A. `(56)/(2505)`
B. `(56)/(6505)`
C. `(56)/(5185)`
D. `(107)/(9605)`

Answer

D

Explanation

Let `S_n=(8)/(5)+(16)/(65)+(24)/(325)+"......"`
`T_(r)=(8r)/(4r^(4)+1)=(8r)/((2^(2)+2r+1)(2^(2)-2r+1))`
`=2[((2^(2)+2r+1)-(2^(2)-2r+1))/((2^(2)+2r+1)(2^(2)-2r+1))]`
`=2[(1)/((2^(2)-2r+1))-(1)/((2^(2)+2r+1))]`
`T_(7)=(8xx7)/(4xx7^(2)+1)=(56)/(9605)`.