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) *2n*1(2n-1)`
` = 8n^(3) -1 - 12n^(2)+6n = 8n^(3) -1^(3)3*2n-1(2n-1)`
` :. S_(n) = SigmaT_(n) = 8 Sigman^(3) - 12 Sigma n^(2) +6Sigma n - Sigma 1 `
` = 8* [ (n(n+1))/2]^(2)-12 xx [ (n(n+1) (2n+1))/6]+6[(n(n+1))/2]-n` ,
`= 2n^(2)(n+1)-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-4n-2 +3]-n`
` = n(n+1)[2n^(2)+2n-4n-2+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) (2n^(2)-1)`