Functional equation $f(x+1)=af(x)+b$ 
Functional equation $f(x+1)=af(x)+b$

There was a question I solved a few days back that asked for a closed form of an equation for a given system. The function came down to this equation which I solved by noting the pattern. By the way $f(0)=10$
So here's how I saw it:
$f(1)=10a+b(1)$
$f(2)=10a^2+b(1+a)$
$f(3)=10a^3+b(1+a+a^2)$
So I saw the pattern and the geometric series in brackets and I managed to figure it out partly because the question format was leading me in that direction.
My question now is, presented purely with a functional equation $f(x+1)=af(x)+b$ for some constants $a, b \in \mathbb R$, and some starting value $f(0)=5$ maybe, would you solve it the way I did or there's a different approach?

$$f(x+1)-af(x)=b~~~~(1)$$
Jet $f(x)=g(x)+c$, then
$$g(x+1)+c-ag(x)-ac=b$$
$$g(x+1)-ag(x)=0, c=b/(1-a)$$
Let $$g(x)=d t^x \implies t=a$$
So the solution of (1) is
$$f(x)=da^x+\frac{b}{1-a}$$
