Answer
$44, 000$
Work Step by Step
Recall the formula:
$\displaystyle \sum_{k=1}^{n}k^{3}=\left[\dfrac{n(n+1)}{2}\right]^{2}$
We can see that for the given sequence, the index does not start at 1. So, we will rewrite the given sequence as:
$\sum_{k=5}^{20} k^{3}=$ (Terms from 5 to 20) = (Terms from 1 to 20) - (Terms from 1 to 4)
We rewrite the sequence as stated above and apply the sum formula:
$\displaystyle \sum_{k=5}^{20} k^{3}= \sum_{k=1}^{20}k^{3}-\sum_{k=1}^{4}k^{3} \\= \displaystyle [\dfrac{20(20+1)}{2}]^{2}-[\dfrac{4(4+1)}{2}]^{2} \\=(210)^{2}-(10)^{2} \\= 44, 000$
