# What is the sum of the first 10 terms of this geometric sequence 3, –6, 12, –24, …?

Jun 27, 2015

$3 , - 6 , 12 , - 24 , \ldots$

You can rewrite this as:

${\left(- 1\right)}^{n} 3 \cdot \left(1 , 2 , 4 , 8 , \ldots\right)$

If we focus on $1 , 2 , 4 , 8. . .$

${a}_{n + 1} = 2 {a}_{n}$
with ${a}_{0} = 1$.

This can be written out as a nice sum:
${\sum}_{n = 0}^{N} {2}^{n} = {2}^{0} + {2}^{1} + {2}^{2} + {2}^{3} + \ldots$
$= 1 + 2 + 4 + 8 + \ldots$

Thus, now we can recombine everything to get:

$\textcolor{g r e e n}{3 {\sum}_{n = 0}^{N} {\left(- 1\right)}^{n} {2}^{n}}$

where $N$ is some arbitrary stop point.

The formula for summing this is:

$r = - 2$
(since ${\left(- 1\right)}^{n} \cdot {2}^{n} = {\left(- 2\right)}^{n}$)

$a = 3$
$n = 10$

$\implies 3 \frac{\left(1 - {\left(- 2\right)}^{10}\right)}{1 + 2}$ $\to {\left(- 2\right)}^{10} = {2}^{10}$

$= 3 \left(\frac{1 - {2}^{5 \cdot 2}}{3}\right)$

$= \cancel{3} \frac{1 - {32}^{2}}{\cancel{3}} = 1 - 1024 = \textcolor{b l u e}{- 1023}$

To sum this up more easily than brute-forcing it, if you don't remember the formula, there's something rather creative you can do.

If you look at the odd-indexed terms ($- 6 , - 24 , - 96 , - 384 ,$ etc), you can see that they are all successively multiplied by $4$. You can see that the even terms ($3 , 12 , 48 , 192 ,$ etc) are also this way. Thus, it lines up like so:

$n = 0 \text{ " " 2 " " " 4 " " " } 6$:

$\text{ } 3 + 12 + 48 + 192 + \ldots$

$n = 1 \text{ " " 3 " " " 5 " " " } 7$:

$- 6 - 24 - 96 - 384 - \ldots$

and you always subtract twice the value of the positive number. Thus, just add up the first five even-indexed numbers (indices $0 , 2 , 4 , \ldots$) and switch the sign.

$= - \left(3 + 12 + 48 + 192 + 768\right)$
$= - 3 \left({4}^{n}\right)$
with ${n}_{0} = 0$

$= - 3 \left({4}^{n}\right) = - 3 \left[{4}^{0} + {4}^{1} + {4}^{2} + {4}^{3} + {4}^{4} + {4}^{5}\right]$
$= - 3 \left(1 + 4 + 16 + 64 + 256\right)$
$= - 3 \left(341\right) = \textcolor{b l u e}{- 1023}$

(You didn't have to write down as many terms here.)

Or, try adding up the original to check:
= 3 - 6 + 12 - 24 + 48 - 96 + 192 - 384 + 768 - 1536)
$= - 3 - 12 - 48 - 192 - 768$
$= - 15 - 240 - 768$
$= - 255 - 768$
$= - 223 - 800 = \textcolor{b l u e}{- 1023}$

(Except now you had to write out 10 terms to add up.)