Unable to prove this by mathematical induction

Question

I have the question:

$$1 + 8 + 27 + \dots + n^3 = \frac {n^2(n+1)^2}n$$

But doesn't this fail the base case? If we plug in a $1$ on both sides of the problem we get $1 = 4$. Which is not true. Does that mean we can't prove this further?

Any help would be extremely appreciated!

Answer 1

The denominator should be $4$. Obviously, the right-hand side needs to be a quartic.

Answer 2

A mathematical proof can be:

I - The problem has a solution, and is determined (or the statement is true);

II - The problem has solutions, and are general and undetermined (or the statement is true for $n$);

III - The problem has no solution (or the statement is false)

Any of the above would be accepted.

You just proved that the statement above is false (III) by assigning $n = 1$.

You could ask your teacher if the intended question is actually wrong (and therefore the denominator on the right side should be 4) or if that was exactly what the question was about, and that is, mathematical proofs of any kind.