algebra precalculus - 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?

Answer

If $T(n) = \frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n}$, then

$$T(n+1) - T(n) = \frac{1}{2^{n+1}}$$
hence if $T(n) = \frac{2^n-1}{2^n}$, then
$$T(n+1) = T(n) + \frac{1}{2^{n+1}} = \frac{2^n-1}{2^n}+\frac{1}{2^{n+1}} = \frac{2(2^n-1)}{2^{n+1}}+\frac{1}{2^{n+1}} = \frac{2^{n+1}-2+1}{2^{n+1}},$$
giving the desired formula.

Since $T(1) = \frac{1}{2}=\frac{2-1}{2}$, the result is established by induction.