Proving Algebraic expression involving a Summation

Question

I'm following a derivation and am stumped by one of the steps:

$\sum_{k=1}^{13}(1 - \frac{k-1}{13})^3 = \frac{1}{13^3}\sum_{k=1}^{13}k^3$

I am stumped as to how

$(14 - (k-1))^3= k^3$

Any help advice appreciated

Thanks Baz

Answer 1

Because it's $$\frac{1}{13^3}\sum_{k=1}^{13}(14-k)^3=\frac{1}{13^3}\sum_{k=1}^{13}k^3$$ Just $$13^3+12^3+...+2^3+1^3=1^3+2^3+...+12^3+13^3.$$

Answer 2

your equation $$\left(1-\frac{k-1}{13}\right)^3=k^3$$ is not true, form here would we get $$1-\frac{k-1}{13}=k$$ and so $$13-k+1=13k$$