SOLUTION: Write the sum without using sigma notation: n over (capital sigma)j=1(under) (-1)^j+1 x^j Thank you very much!

Question 48744This question is from textbook College Algebra
: Write the sum without using sigma notation:
n over (capital sigma)j=1(under) (-1)^j+1 x^j
Thank you very much! This question is from textbook College Algebra

Answer by kev82(151) (Show Source): You can put this solution on YOUR website!
Hi,
So we need to find a formula for

To figure out what's actually happenning, lets write out a few terms (I'm only going to work with even N for now)

Or splitting this into two sequences.

You should be able to see that the first sequence is just a factor of -x different from the first.
The sequence clearly has N/2 terms(remember I'm assuming N even at moment)
Using the formula for a geometric series (1-x^n)/(1-x) I'm sure you've seen it, we can see that the sequence we want

where

is given by

Using this and the fact we want

and

copies of this sequence we get the general formula (lots of tidying up)

However, we haven't dealt with the n=odd case. If you try out a few examples you will note that when n is odd there is an extra term

So we need to add this when n is odd. The easiest way to do this is with a function that is 1 for odd and 0 for even.
Adding

gives the final answer of

It's quite possible I've made a mistake, but I've checked it with a few cases and it works so I'm pretty confident.
Of course the geometric formula isn't valid for x=1, so neither is this. I'll let you solve that case your self - it's pretty simple though.
Hope that helps,
Kev
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