Of course, that is not the complete proof.
The formula has been derived by examining a pattern from the first few terms of n.
To be sure whether this formula holds for all n natural numbers,
we try to establish that the formula being true for some n=k,
causes the formula to be true for the next n=k+1.
Because we use n or k, we are attempting to establish this term-by-term relationship in general
for all pairs of terms.
If it works then a chain-reaction is established for all n.
To do this we try to prove
Therefore, we start by adding the next cube
Therefore the formula holds for all n, as it holds for n=1.