Question 114
Let $(a_n)_{n=1}^\infty$ be a sequence of positive numbers defined with: $a_1 = 1$, $a_2 = 2$ and $\frac{a_{n+1}^4}{a_n^3} = 2a_{n+2} - a_{n+1}$. Prove that for all positive integers $N \gt 1$ it holds that
$$
\sum_{k=1}^{N} \frac{a_k^2}{a_{k+1}} \lt 3.
$$
$$
\sum_{k=1}^{N} \frac{a_k^2}{a_{k+1}} \lt 3.
$$