# How do you find the sum of the infinite geometric series 1-2+4-..+1024?

Dec 2, 2015

Apply the geometric series formula to find
$1 - 2 + 4 - \ldots + 1024 = 683$

#### Explanation:

Given a geometric series with initial value $a$ and common ratio $r$, we have the formula

${\sum}_{k = 0}^{n - 1} a {r}^{k} = a \frac{1 - {r}^{n}}{1 - r}$

(See What is the formula for the sum of a geometric sequence? for a proof)

In our exercise, the first term is clearly $a = 1$ and we can find the common ratio by dividing the second term by the first to obtain
$r = \frac{- 2}{1} = - 2$

Finally, we can find that $n = 11$ by noting that
$1024 = {\left(- 2\right)}^{10} = {r}^{10}$

$\implies 1 - 2 + 4 - \ldots + 1024 = {\sum}_{k = 0}^{11 - 1} 1 \cdot {\left(- 2\right)}^{k}$

Thus, applying the formula gives us

${\sum}_{k = 0}^{11 - 1} 1 \cdot {\left(- 2\right)}^{k} = 1 \frac{1 - {\left(- 2\right)}^{11}}{1 - \left(- 2\right)} = \frac{2049}{3} = 683$