Question

How do you evaluate `\sum _ { n = 1} ^ { 20} ( n + n ^ { 3} )`?

Answer 1

`sum_(n=1)^20 (n+n^3) = 44310`

Explanation:

Let:

`S = sum_(n=1)^20 (n+n^3)`

We need the standard formula:

`sum_(r=1)^n r \ = 1/2n(n+1)`
`sum_(r=1)^n r^2 = 1/4n^2(n+1)^2`

Which gives us:
`S = sum_(n=1)^20 n+sum_(n=1)^20 n^3`
`\ \ = 1/2(20)(21)+1/4(20^2)(21^2)`
`\ \ = 1/2(420)+1/4(400)(441)`
`\ \ = 210 + (100)(441)`
`\ \ = 210 + 44100`
`\ \ = 44310`