If the sum to n terms of the series `(1)/(1*3*5*7)+(1)/(3*5*7*9)+(1)/(5*7*9*11)+"......"` is `(1)/(90)-(lambda)/(f(n))`, then find `f(0), f(1)` and `f(lambda)`

Options

A. `f(0)=15`
B. `f(1)=105`
C. `f(lambda)=(640)/(27)`
D. `lambda=(1)/(3)`

Answer

A, B, C

Explanation

`T_(n)=(1)/((2n-1)(2n+1)(2n+3)(2n+5))`
`:.S_(n)=sum_(n=1)^(n)T_(n)a`
`S_(n)=(1)/(6)sum_(n=1)^(n)((2n+5)-(2n-1))/((2n-1)(2n+1)(2n+3)(2n+5))`
`=(1)/(6)sum_(n=1)^(n)((1)/((2n-1)(2n+1)(2n+3))-(1)/((2n+1)(2n+3)(2n+5)))`
`=(1)/(6)((1)/(1*3*5)-(1)/((2n+1)(2n+3)(2n+5)))`
`=(1)/(90)-((1)/(6))/((2n+1)(2n+3)(2n+5))`
`:.lambda=(1)/(6)`
and `f(n)=(2n+1)(2n+3)(2n+5)`
`:.f(0)=15`
`f(1)=105`
and `f(lambda)=f"((1)/(6))((1)/(3)+1)((1)/(3)+3)((1)/(3)+5)=(640)/(27)`.