Let $a_n;\;n> 1$ be a sequence of positive numbers such that $a_1, a_2, a_3$ are in $AP$, $a_2, a_3, a_4$ are in $GP$, $a_3, a_4, a_5$ are in $AP$, $a_4, a_5, a_6$ are in $GP$, and so on. Find an expression for $a_n$ in terms of $a_1$ and $a_2$.
I did
$a_3=2a_2+a_1$
$a_4=a_2+a_1\bigg(\frac{1}{a_2}-4\bigg)$
$a_5=6a_2+a_1\bigg(\frac{2}{a_2}-7\bigg)$
But I can't see how to write $a_n$ in terms of $a_1$ and $a_2$... Can somenone help me? Thanks for attention!
Hint:
Apply induction to prove: $$ a_n=\frac1{a_2}\times\begin{cases} \left(\frac{n+1}2a_2-\frac{n-1}2a_1\right)\left(\frac{n-1}2a_2-\frac{n-3}2a_1\right),& n\text{ odd},\\ \left(\frac{n}2a_2-\frac{n-2}2a_1\right)^2,& n\text{ even}. \end{cases} $$