I see the geometric series stated in two ways, in text books and other sources on the internet.

Generally an infinite geometric series has the form

and a finite form of the geometric series generally is as such

Now for example, even thought I've found my own way of solving this problem, when I come to others that involve taking a finite form of the partial sum, I am unsure if my method is good enough, even though it is correct 90% of the time, I want to be accurate.

Example:

Here is the infinite series of

Now here is the finite form of the series of to the 20th term

The finite form of the summation was written as such, but they state the general form of a finitie series is to the n-1 power. So I would rewrite this as, if I followed the form that was mentioned

Now If I follow this, the first term I get is

Following the form for a finite series partial sum is

However, this is not quite right as provided by the book or other solutions I've seen my friends do of this problem, when I do the way I found is more accurate for me

Where I look at the finite series partial sum form as this, the first term unbroken to help simplifying exponents

Where

is the first term in the series and

is the last term in the series + 1, works the same for the general case of the nth term

Similarily

Which is the correct term before further simplifiction, so whats the deal with the form they mention and other sources I have seen explaining the geometric series? Confuses me at times. Sorry for the lengthy post, I know how MHF want poster to include all information