# Finding intial conditions for closed sequences

I am having trouble finding initial conditions for closed sequences. Could you help me get through this problem from my textbook? Thanks!

Show that $3 · 2^n + 7 · 5^n$ is a solution to the recurrence relation, $a_{n} = 7a_{n−1} − 10a_{n−2}$. What would the initial conditions need to be for this to be the closed formula for the sequence?

• Can you not simply plug in $n = 0, 1, 2$ into the expression? – Brian Tung Mar 22 '18 at 19:43
• Yes thank you @BrianTung – Ethan Walser Mar 22 '18 at 19:48

The initial conditions are $a_0=3+7=10$ and $a_1=3\times2+7\times5=41$, of course.
• That means $a_n = 7a_{n−1} − 10a_{n−2}$? – Mathew Mahindaratne Mar 22 '18 at 19:47
• @MathewMahindaratne The only sequence that satisfies the initial conditions that I mentioned and such that $a_n=7a_{n-1}-10a_{n-2}$ is the one mentioned by the OP. – José Carlos Santos Mar 22 '18 at 19:49