<s>[INST] How can I show using mathematical induction that $\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \frac{2^n - 1}{2^n}$ How can I show using mathematical induction that $\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \frac{2^n - 1}{2^n}$
Edit: I'm specifically stuck on showing that $\frac{2^n - 1}{2^n} + \frac{1}{2^{n+1}} = \frac{2^{n+1}-1}{2^{n+1}}$.
What should the approach be here?
 [/INST] For what is worth:
$$S_n = \frac{1}{2} + \frac{1}{4} +  \cdots  + \frac{1}{{{2^n}}}$$
Then
$$\frac{{{S_n}}}{2} = {S_n} + \frac{1}{{{2^{n + 1}}}} - \frac{1}{2} \Rightarrow {S_n} = 2{S_n} + \frac{1}{{{2^n}}} - 1 \Rightarrow   1 - \frac{1}{{{2^n}}} = {S_n}$$
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