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.
a0 = 4
a3 = 6
So
a2 + 2a1 = 6 (1)
a1 + 2a0 = a2 → a1 + 2*4 = a2 → a1 - a2 = -8 (2)
Add (1) and (2)
3a1 = -2
a1 = -2/3
a2 = 6 - 2a1 = 6 - (2)(-2/3) = 22/3
a4 = a3 + 2a2
a4 = 6 + 2(22/3) = 62/3
a5 = 62/3 + 2(6) = 98/3
0 1 2 3 4 5
4 -2/3 22/3 6 62/3 98/3
