Find the sum to n terms of the series :`1^2+(1^2+2^2)+(1^2+2^2+3^2)+dotdotdot`

Answer

N/a

Explanation

nth term
`T_(n) = 1^(2) + 2^(2) + 3^(2) + … + n^(2)`
`= (1)/(6)n(n + 1)(2n + 1) = (1)/(6)(2n^(3) + 3n^(2) + n)`
`rArr` `S_(n) = (1)/(6)[2 sum n^(3) + 3 sum n^(2) + sum n]`
`= (1)/(6)[(2)/(4)n^(2)(n + 1)^(2) + (3)/(6)n(n + 1)(2n + 1) + (1)/(2)n (n + 1)]`
`= (1)/(12)n (n + 1) [n(n + 1) + (2n + 1) + 1]`
`= (1)/(12)n(n + 1)[n(n + 1) + (2n + 2)]`
`= (1)/(12)n(n + 1) [n(n + 1) + 2 (n + 1)]`
`= (1)/(12)n (n + 1)^(2)(n + 2)`