sequences

The sequence $a_n$ has the property that $a_n = a_{n-1} + 2a_{n-2}$ for $n \ge 2$. It is also true that $a_0 = 4$ and $a_3 = 6$. What is the value of $a_5$? Express your answer as a common fraction.

For n = 3 and n = 2:

$a_3 = a_2 + 2 a_1\\ a_2 = a_1 + 2a_0$

substituting,

$6=a_2+2a_1\\ a_2 = a_1+8\\ $

this is a simple system of equations.

we can find $a_n$ for any n now by plugging $a_{n-1}$ and $a_{n-2}$ into the formula.

For example, $a_4 = a_3 + 2a_2$ (which we know already)