I'm trying to teach myself mathematics. I only started last year with a previous knowledge of basic arithmetic. I often make a detour into regions featuring problems that are a bit too deep for me. I'm often too curious to wait until I was able to catch up on the necessary bedding. I hope you'll excuse me :-)
$$f(x) = f(x-1)*b \to f(x) = c_1 b^{x-1}$$
This is how I would compute the original recursive algorithm(?):
$$f(x) = f(x-1)*b$$
$$\begin{align} \\ f(4) &= f(4-1)*b \\ &= f(f(3-1)*b)*b \\ &= f(f(f(2-1)*b)*b)*b \\ &= f(f(f(f(1-1)*b)*b)*b)*b \\ &= \qquad ... \end{align}$$
But how do I compute the recurrence relation $f(x) = c_1 b^{x-1}$?
I would compute the Fibonacci numbers like this:
$$\begin{align} F_n &= F_{n-1} + F_{n-2}\\ F_0 &= 0\\ F_1 &= 1\\ F_5 &= F_{5-1} + F_{5-2}\\ &= F_4 + F_3\\ &= (F_3 + F_2) + (F_2 + F_1)\\ &= ((F_2 + F_1) + (F_1 + F_0)) + ((F_1 + F_0) + F_1)\\ &= (((F_1 + F_0) + F_1) + (F_1 + F_0)) + ((F_1 + F_0) + F_1) \\ &= 5 \end{align}$$
I'm also not sure about the difference between a recursive algorithm and its recurrence relation. I'd be grateful if someone could enlighten me or simply link me up.