Finding the Sum of an Alternating Geometric Sequence

[Sarah00]

Hi!

If I have a sequence that its first 4 terms are:

3⁰, -3¹, +3², -3²

The pattern is geometric sequence but has alternating signs..

How can I find its sum ..

I know it is composed of 2 sequences ..

However, when I try to separate the 2 sequences .. I get them of different "lengths"

In other words, it is (3⁰+3²+3⁴) - (3¹+3³) for 5 terms

but for 4 terms I get:
In other words, it is (3⁰+3²) - (3¹+3³)How can I get general formula for both ..

(-1)^k helps ! but how! and what about the number of termsThanks!

 

[Sarah00]

Further to my previous post ..

 

[Sarah00]

This is the book's answer:

 

[Ssnow]

When $n$ is odd that is $n=2m+1$ you obtain $\frac{1}{4}(-3^{2m+1}+1)$, when is even $n=2m$ you obtain $\frac{1}{4}(3^{2m+1}+1)$. In order to obtain both you must have $\frac{1}{4}((-1)^{n}3^{n+1}+1)$ (this is obtained merging the two previous...)

 

[PeroK]

Further to my previous post ..

If you set $n = 1$, then your formula gives $S = -1/2$