Finding the $n$th term in a 2nd order linear recursive sequence

Question

$\begin{gathered}a_{n}=a_{n+1}+a_{n+2}\\ a_{1}=3\\ a_{5}=55\\ a_{{}6}=?\\ \end{gathered}$

I've tried solving it through the Fibonacci sequence, but it didn't work either. I am required to find the sixth term in this sequence (recursive). Help appreciated.

Answer 1

$$a_1=a_2+a_3$$$$=2a_3+a_4$$$$=2(a_4+a_5)+a_4$$$$=3a_4+2a_5$$$$=3(a_5+a_6)+2a_5$$$$=5a_5+3a_6$$ As per question,$$\therefore a_1=5a_5+3a_6$$$$\implies a_6=\frac{3-5(55)}{3}=-90.67$$