A geometric series is a list of numbers where each number, or term, is found by multiplying the previous term by a common ratio r. If we call the first term a, then the geometric series can be expressed as follows:
We call this a finite geometric series because there is a limited number of terms (an infinite geometric series continues on forever.) In this example, there are 10 terms, the common ratio is r, and each of the terms of the geometric sequence follows the same pattern. The first term is a. The second term is the previous term a multiplied by r. The third term is the second term ar multiplied again by r to create , and so on until the last term.
We can represent the sum of the first n terms of a finite geometric series with this equation:
You might see this common sigma notation, which is shorthand to express the summation of a list of consecutive terms. Instead of listing all the n terms individually, you see one general term (represented by ) and the range of terms you can generate by substituting incremental values into that general term.
The lower expression k=0 is where you start, and the top number is where you end. Each term is found by replacing n in the expression to the right of the sigma. Sigma notation can be used to represent the sum of a finite geometric series, the sum of an infinite geometric series, or the sum of other kinds of series as well.
For example, sigma notation for summing up the first 10 terms of a finite geometric series can be shown as:
You’ll notice this is the same as above when you simply to a and to ar.
Summing Up the Finite Geometric Series
The sum of the first n terms can be found using this formula:
In sigma notation, you could write this out as:
Don't confuse the numerator and the denominator here. The numerator includes an exponent equivalent to the number of terms you are summing up.
Let’s look at how this works. Say we have a finite geometric series: 5, 10, 20, 40, 80…
The common ratio r here is 2.
The first term a is 5.
The fourth term is
To find the sum of the first 7 terms, we would use the equation:
When substituting the terms we identified, n = 7 , r = 2, and a = 5, we get:
We can check our answer the manual way:
The Sum of a Finite Geometric Series Made Simple
You can use sigma notation to express the summation of a finite number of terms (though in higher math, you will see how it works even for infinite numbers).
Keep this sum formula for the first n terms of a geometric series handy:
If you understand this equation, you’ll be on your way to summing up finite geometric series in no time.