Sum of n terms of the following series`1^3+3^3+5^3+7^3+`

Options

A. `n^(2)(2n^(2)-1)`
B. `n^(3)(n-1)`
C. `n^(3)+8n+4`
D. `2n^(4)+3n^(2)`

Answer

A

Explanation

`T_(n)=(2n-1)^(3)`
`=8n^(3)-1^(3)-3*2n*1(2n-1)`
`=8n^(3)-1-12n^(2)+6n`
`=8n^(3)-12n^(2)+6n-1`
`thereforeS_(n)=sumT_(n)`
`=8sumn^(3)-12sumn^(2)+6sumn-sum1`
`=8*[(n(n+1))/(2)]^(2)-12xx[(n(n+1)(2n+1))/(6)]+6[(n(n+1))/(2)]-n`
`=2n^(2)(n+1)^(2)-2n(n+1)(2n+1)+3n(n+1)-n`
`=n(n+1)[2n(n+1)-2(2n+1)=3]-n`
`=n(n+1)[2n^(2)-2n+1]-n`
`=n(n+1)*2n(n-1)+n(n+1)-n`
`=2n^(2)(n^(2)-1)+n^(2)`
`=n^(2)(2n^(2)-1)`