Solving simultaneous recurrence relations.

Question

Need help solving the following system: $$s_n=s_{n-1}+2t_{n-1}+2$$ $$t_n=-2s_{n-1}+t_{n-1}+n$$ $$n\ge2$$ $$s_1=1$$ $$t_1=2$$

Answer 1

Hint

$$s_n=s_{n-1}+2t_{n-1}+2\quad (1)\\ t_n=-2s_{n-1}+t_{n-1}+n\quad (2)$$

make $-2\cdot(2)+(1)$:

$$s_n-2t_n=5s_{n-1}+2(1-n)\to t_n=\frac{s_n-5s_{n-1}-2(1-n)}{2}$$

replacing at $(1)$:

$$s_n=s_{n-1}+2t_{n-1}+2\to s_n=s_{n-1}+s_{n-1}-5s_{n-2}-2(2-n)+2$$

$$s_n-2s_{n-1}+5s_{n-2}+2(1-n)=0$$

Can you finish?