When your pre-calculus 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.