sequences and series - finding $a_1$ in an arithmetic progression

Given an arithmetic progression such that:

$$a_{n+1}=\frac{9n^2-21n+10}{a_n}$$

How can I find the value of $a_1$?

I tried using $a_{n+1}=a_1+nd$ but I think it's a loop..

Thanks.

Answer

We have

$$a_2=\frac{9-21+10}{a_1}\Rightarrow a_1a_2=-2\tag1$$
and
$$a_3=\frac{36-42+10}{a_2}\Rightarrow a_2a_3=4\tag2$$

Since we have $a_1+a_3=2a_2$, with $(1)(2)$, we have

$$a_1+\frac{4}{a_2}=2a_2\Rightarrow a_1a_2+4=2a_2^2\Rightarrow a_2=\pm1.$$
Can you take it from here?