<s>[INST] Question abut the ambiguity of a maths problem. I am trying to prove by induction that:
$$1^3-2^3 +\cdots+n^3=(1+2+\cdots+n)^2 $$
This was a problem from a practice worksheet, but I don't understand how to interpret the LHS.
Is the following correct:
$$1^3-2^3+3^3-4^3+5^3-6^3+\cdots+n^3$$
Or this:
$$1^3-2^3+3^3+4^3+5^3+6^3+\cdots+n^3$$
I have a feeling it is the former. If this is the case, I presumably have to consider odd and even cases for n right?
Thanks
 [/INST] The true formula is that
$1^3+2^3+...+n^3=(1+2+...+n)^2$,
and you can prove this by induction. Assume that
$1^3+2^3+\cdots +(n-1)^3=(1+2+\cdots +(n-1))^2$,
then
$1^3+2^3+\cdots +n^3=(1+2+...+n-1)^2+n^3=\frac{n^2(n-1)^2}{4}+n^3=(1+2+\cdots n)^2.$
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