I have a formula for a series which looks like this:
$\sum_{i=2}^{8}(-2)^i$
And I see that this series gives me the following output:
4 – 8 + 16 – 32 + 64 – 128 + 256
Which gives a sum of 172 – assuming I have done the calculations correctly.
However, I also need to find a formula to determine the sum, and I'm wondering if I'm on the right path here:
$\sum_{i=0}^{7}2^i = \frac{2^{7^+1}-1}{2-1} $
This is a formula I found to determine the sum of a geometric series. But I'm a bit thrown off by the task I'm trying to solve above.
I tried approaching the formula as shown below:
$ \frac{(-2)^{6}-1}{-2-1} $
But seing as the formula I have been working on above is a negative integer I'm having some difficulties finding the right approach to calculate the sum with the correct formula.
Am I using the wrong formula or am I simply just using it incorrectly?
Best Answer
We have that for any $r\neq 1$
$$\sum_{i=0}^{n}r^i = \frac{r^{n+1}-1}{r-1}$$
then
$$\sum_{i=0}^{8}(-2)^i = \frac{(-2)^{9}-1}{-2-1}=171$$
and
$$\sum_{i=2}^{8}(-2)^i =\sum_{i=0}^{8}(-2)^i-\sum_{i=0}^{1}(-2)^i=171-(1-2)=172$$