Show by induction that the following formulas hold for
$$ \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4} $$
Not sure how to go about this problem. Can someone help please? Thanks
Show by induction that the following formulas hold for
$$ \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4} $$
Not sure how to go about this problem. Can someone help please? Thanks
The base case: $$\sum_{i=1}^1 i^3 = 1^3 = \frac{1^2 2^2}{4}.$$ Now suppose that the result holds for all $k \leq n$. Then \begin{align} \sum_{i=1}^{n+1} i^3 &= \sum_{i=1}^n i^3 + (n+1)^3 \\ &= \frac{n^2 (n+1)^2}{4} + (n+1)^3 \\ &= (n+1)^2 \left( \frac{n^2 }{4} + (n+1) \right) \\ &= \frac{(n+1)^2}{4}(n^2 + 4n + 4) \\ &= \frac{(n+1)^2}{4}(n+2)^2 \\ &= \frac{(n+1)^2(n+2)^2}{4}. \end{align}