Im trying to write recursive formulas for sequences but it seems like there are different techniques depending on what type of sequence I'm dealing with. for example I want to the sequence:
$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} +.... \frac{1}{n}$
What I see when I look at this is all the next terms are $\frac{1}{2}$ the previous term. So, I wrote the following recursive formula - which is incorrect.
$f(1) = 1, f(n) = f(n-1) + \frac{f(n-1)}{2}$ Why is this wrong?
Why is this correct? $f(1) = 1, f(n) = f(n-1) +\frac{1}{2}^{n-1}$
Intuitively I thought that my original attempt looked correct.