How can I convert sigma notation into a finite geometric series?

[A.J.710]

Homework Statement

http://postimage.org/][/PLAIN]
image url

Homework Equations

N/A

The Attempt at a Solution

I have to solve a different question using a similar method to this one but I cannot figure out how they got the sigma notation into the last equation format. I tried writing out the summation manually as well as the equation substituting n for 0,1,2,3 but they don't match up and I cannot do my next problem without figuring this one out first.

 

[SteamKing]

Homework Statement

[/PLAIN]
image url

Homework Equations

N/A

The Attempt at a Solution

I have to solve a different question using a similar method to this one but I cannot figure out how they got the sigma notation into the last equation format. I tried writing out the summation manually as well as the equation substituting n for 0,1,2,3 but they don't match up and I cannot do my next problem without figuring this one out first.

A little context would be helpful.

What are α and β? What does u[n] represent?

 

[A.J.710]

A little context would be helpful.

What are α and β? What does u[n] represent?

α and β are just variables, there is no value to them in the question. u[n] is a unit step function basically making the function true at n > 0 and 0 at n < 0.

The original question is a convolution question x[n] * h[n] in linear systems but what is in the picture is all that is really involved. There is nothing else too it I could post.

 

[Mark44]

In the summation, you have $(\frac{\alpha}{\beta})^0 + (\frac{\alpha}{\beta})^1 + (\frac{\alpha}{\beta})^2 + \dots + (\frac{\alpha}{\beta})^n$, which is a finite geometric series with n + 1 terms. There's a formula for the sum of a geometric series.