Geometric progression, is this correct? ratio/first/next term

[mr_coffee]

Hello everyone, I'm trying to solve the following:
If n is an integer and n > = 1, find a formula for the expression
$$2^n - 2^{n-1} + 2^{n-2} - 2^{n-3} + ... + (-1)^{n-1} * 2 + (-1)^n$$

okay this confuses me, because I'm not sure which is the first term and which is the last term...
I figured the ratio was the following: -2
because if u take 2^n/[-2^(n-1)] = -2
I took the first term and divided it by the 2nd term to find the ratio or is it vice versa? by taking the 2nd term and dividing it by the first, which would give the ratio of: -1/2?

once i find the ratio, i can find the term right after the last by multiplying the ratio by the last term in this case (-1)^n is the last term, and the first term is 2^n correct?

Thanks, once i get this figured out i cna find the formual for the sum.

 

[courtrigrad]

You take the 2nd term and divide it by the first, so $$r = -\frac{1}{2}$$

 

[mr_coffee]

Thanks courtrigrad
Okay so the answer I'm getting would be:
["Mythical Next term" - "First real term" ]/[ratio - 1]
thats the general formula to find the formula for the sum of a geometric progression and i come up with:

sum = [-.5*(-1)^n - 2^n]/(-3/2)

does that look correct to you? I'm not sure how i can clean that up though.

 

[courtrigrad]

$$\sum_{n=0}^{\infty} ar^{k} = \frac{a(1-r^{n+1})}{1-r}$$

The last term is $$(-1)^{n-1} 2^{n}$$

$$\frac{2^{n}(1-(-\frac{1}{2})^{n+1}))}{\frac{3}{2}}$$

 

[mr_coffee]

thanks a ton! I now see that the (-1)^n was just changing the + to - and vice versa! :)