<s>[INST] Find a pattern and prove it by mathematical induction: $1 = 1$
$3 + 5 = 8$
$7 + 9 + 11 = 27$
$13 + 15 + 17 + 19 = 64$
Etc...

I am having trouble seeing a pattern with this, I know it is relatable with Fibonacci Numbers but I am having trouble grasping this topic
 [/INST] Let $a_n$ the $n$-th term of the sequence.
The sum of the $n$ first terms is the sum of the first $\frac{n(n+1)}{2}$ odd integers, which is $\left( \frac{n(n+1)}{2} \right)^2$.
So that:
\begin{equation}
a_1+\ldots +a_n = \left( \frac{n(n+1)}{2} \right)^2
\end{equation}
\begin{equation}
a_1+\ldots +a_{n-1} = \left( \frac{(n-1)n}{2} \right)^2
\end{equation}
Then
$$a_n = \left( \frac{n(n+1)}{2} \right)^2 - \left( \frac{(n-1)n}{2} \right)^2 = \\
\left( \frac{n(n+1)}{2}+\frac{(n-1)n}{2} \right)\left( \frac{n(n+1)}{2}-\frac{(n-1)n}{2} \right) = 
\left( \frac{n^2+n}{2}+\frac{n^2-n}{2} \right)\left( \frac{n^2+n}{2}-\frac{n^2-n}{2} \right) = 
n^2 \cdot n = n^3
$$
</s>