Find the sum of the series `(1^3)/1+(1^3+2^3)/(1+3)+(1^3+2^3+3^3)/(1+3+5)+` up to `n` terms.

Options

A. `(1)/(8)n(2n^(2) + 9n + 13)`
B. `(1)/(24)n(2n^(2) + 9n + 13)`
C. `(1)/(4)(2n^(2) + 9n + 13)`
D. `(1)/(2)n(2n^(2) + 9n + 13)`

Answer

`B`

Explanation

`(1^(3))/(1) + (1^(3) + 2^(3))/(1 + 3) + (1^(3) + 2^(3) + 3^(3))/(1 + 3 + 5) + ...`
`T_(n) = (1^(3) + 2^(3) + 3^(3) + ..."to n terms")/(1 + 3 + 5 + ..."to n terms")`
`= (sumn^(3))/((n)/(2)[2(1) + (n - 1)2])`
`= ((1)/(4)n^(2)(n + 1)^(2))/(n^(2)) = (1)/(4)(n^(2) + 2n + 1)`
`rArr` `S_(n) = (1)/(4)[sumn^(2) + 2sumn + sum1]`
`= (1)/(4)[(1)/(6)n(n + 1)(2n + 1) + (2)/(2)n(n + 1)+n]`
`= (1)/(24)n[(n + 1)(2n + 1) + 6(n + 1)+6]`
`= (1)/(24)n(2n^(2) + 3n + 1 + 6n + 6 + 6)`
`= (1)/(24)n(2n^(2) + 9n + 13)`