<s>[INST] Finding an exact solution to a difference equation How would I solve an equation of the form:
$u(n+1)=1/2u(n)+(1/3)^n$ when $u(0)=1$?
I got an answer of the form $u(n)= c + \sum(1/3)^j*2^{j-1}$ but believe this is incorrect?
 [/INST] Consider $u_{n+1} = \dfrac{1}{2}\cdot u_n$. This gives: $u_n = 2^{-n}$, $n\geq 0$. So the general solution would be: $u_n = a\cdot 2^{-n} + b\cdot 3^{-n}$. 
$u_0 = 1 \to 1 = a + b$, and $u_1 = 0.5u_0 + 1 = \dfrac{3}{2} \to \dfrac{a}{2} + \dfrac{b}{3} = \dfrac{3}{2} \to 3a + 2b = 9 \to 3a + 2(1 - a) = 9 \to a = 7$, and $b = 1 - a = 1 - 7 = -6$. Thus:
$u_n = 7\cdot 2^{-n} - 6\cdot 3^{-n}$, $n \geq 0$
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