How to Find the Partial Sum of a Geometric Sequence
When your precalculus teacher asks you to find the partial sum of a geometric sequence, the sum will have an upper limit and a lower limit. The common ratio of partial sums of this type has no specific restrictions.
You can find the partial sum of a geometric sequence, which has the general explicit expression of
by using the following formula:
For example, to find
follow these steps:

Find a_{1} by plugging in 1 for n.

Find a_{2} by plugging in 2 for n.

Divide a_{2} by a_{1} to find r.
For this example, r = –3/9 = –1/3. Notice that this value is the same as the fraction in the parentheses.
You may have noticed that 9(–1/3)^{n}^{ – 1} follows the general formula for
(the general formula for a geometric sequence) exactly, where a_{1} = 9 and r = –1/3. However, if you didn’t notice it, the method used in Steps 1–3 works to a tee.

Plug a_{1}, r, and k into the sum formula.
The problem now boils down to the following simplifications:
Geometric summation problems take quite a bit of work with fractions, so make sure to find a common denominator, invert, and multiply when necessary. Or you can use a calculator and then reconvert to a fraction. Just be careful to use correct parentheses when entering the numbers.