The sequence a_n has the property that a_n = a_{n-1} + 2a_{n-2} for n >= 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.
+505 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)
