Question

Find the general solution of the recurrence relation:

 an = an-1 + 2an-2 ,  with a0 = 2 and a1 = 7.

Answer 1

Let us find the solution of the recurrence relation $a_n = a_{n-1} + 2a_{n-2}$ ,  with $a_0 = 2$ and $a_1 = 7$. Let us solve the characteristic equation $k^2=k+2$ which is equivalent to $k^2-k-2=0$, and hence by Vieta's formulas has the solutions $k_1=-1$ and $k_2=2.$ It follows that the solution of the equation is $a_n=C_1\cdot (-1)^n+C_2\cdot 2^n.$ Since $a_0 = 2$ and $a_1 = 7$, we have that $2=a_0=C_1+C_2$ and $7=-C_1+2C_2.$ Therefore, $C_2=3$ and $C_1=-1.$ We conclude that $a_n=(-1)^{n+1}+3\cdot 2^n.$