Find $a_{n}$ if $(a_{n})$ is a sequence such that $a_{1} := 1$ and $\frac{1}{a_{n+1}} = \frac{2}{a_{n}} + 3$ for $n \geq 2$?

Question

This problem is weird. By the initial condition $a_{1} = 1$ we have $a_{2} = \frac{1}{5}$ and so on. But is there really a pattern for $a_{n}$?

I guess this problem is that kind of problems that require some luck to see the solution?

Answer 1

HINT:

Let $1/a_n=b_n+cn+d\implies b_1=\cdots$

So, we have $b_{n+1}+c(n+1)+d=2\left(b_n+cn+d\right)+3$

$\iff b_{n+1}=2b_n+cn+d+3$

Set $d+3=0,c=0$