Answer
$$TRUE.$$
Work Step by Step
If we prove that for every positive integer $n$
\[
1^{3}+2^{3}+\ldots+n^{3}=(1+2+\ldots+n)^{2}
\]
is satisfied, then the statement is true.
For $n=1: 1^{3}=1=1^{2}:$ thus, the statement is true for $1=n$
For $n>1:$ We rewrite left side as
\[
1^{3}+2^{3}+\ldots+n^{3}=\sum_{k=1}^{k=n} k^{3}
\]
Using $[\text { Theorem5.4.2 }(c)$ ) we get:
\[
\sum_{k=1}^{k=n} k^{3}=\left[\frac{(1+n) \cdot n}{2}\right]^{2}
\]
We use ${\text { Theorem5.4.2 }(a)}$
\[
\left[\frac{(1+n)\cdot n }{2}\right]^{2}=\left[\sum_{k=1}^{k=n} k\right]^{2}=[1+2+\ldots+n]^{2}
\]
Thus, from above we obtain
\[
1^{3}+2^{3}+\ldots+n^{3}=(1+2+\ldots+n)^{2}
\]
Thus, the statement is TRUE
