Question

Find a formula for
1/2 + 1/4 + 1/8 + ... + 1/(2^n)
by examining the values of this expression for small n. Prove your
formula.

Answer 1

Do by induction

Answer 2

I can prove it, but I do not know how to find the formula itself.

Answer 3

The formula is
\[\sum_{k=m}^{n} r^k=\frac{r^m-r^{n+1}}{1-r}\]

So r=2, m=1 and n remains as n so


\[\sum_{k=1}^{n} (\frac{1}{2})^k=\frac{(\frac{1}{2})^1-(\frac{1}{2})^{n+1}}{1-(\frac{1}{2})}\]

simplify



\[\sum_{k=1}^{n} (\frac{1}{2})^k = 1-(\frac{1}{2})^{n}\]

This is a Geometric Series.