`S_(n)` be the sum of n terms of the series `(8)/(5)+(16)/(65)+(24)/(325)+"......"`
The value of `S_(8)`, is (a)`(288)/(145)` (b)`(1088)/(545)` (c)`(81)/(41)` (d)`(107)/(245)`

Options

A. `(288)/(145)`
B. `(1088)/(545)`
C. `(81)/(41)`
D. `(107)/(245)`

Answer

A

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))]`
`S_(8)=sum_(r=1)^(8)T_(r)=2sum_(r=1)^(8)((1)/(2^(2)-2r+1)-(1)/(2^(2)+2r+1))`
`=2(1-(1)/(2(8)^(2)+2(8)+1))=2(1-(1)/(145))=(288)/(145)`