A **finite geometric sequence **is a list of numbers (terms) with an ending; each term is multiplied by the same amount (called a common ratio) to get the next term in the sequence.

For example: the sequence 5, 10, 20, 40, 80, … 320 ends at 320. Each term is multiplied by 2 to get the next term.

**Note**: A slightly different form is the geometric series, where terms are added instead of listed: *a + ar + ar ^{2} + ar^{3} + …*. These behave differently, and their sums are different. This article is about the geometric

**sequence**; If you want to learn about the

**series**, see: What is a Geometric Series?

## Nth Term of a Geometric Sequence

The general form of the sequence is a_{1}, a_{1}r_{2}, a_{1}r_{3}, a_{1}r_{4},… a_{1}r_{(n – 1)}

.We don’t always want to write out the entire sequence all the time, so instead of writing everything out (5, 10, 20, 40, 80, 160…) we can use a much shorter formula. The general **formula for the nth term of a geometric sequence ** is:

a_{n}= a_{1}r^{(n – 1)}

Where:

- a
_{1}= the first term in the sequence, - r = the common ratio.
- n = the nth term.

For the example sequence above, the common ratio is 2 and the first term is 5. We can find out the nth terms by plugging those into the formula:

**a _{n} = 5 · 2^{(n – 1)}**.

- First term: 5 · 2
^{(1 – 1)}= 5 · 2^{(0)}= 5 · 1 = 5 - Second term: 5 · 2
^{(2 – 1)}= 5 · 2^{(1)}= 5 · 2 = 10 - Third term: 5 · 2
^{(3 – 1)}= 5 · 2^{(2)}= 5 · 4 = 20 - Fourth term: 5 · 2
^{(4 – 1)}= 5 · 2^{(3)}= 5 · 8 = 40

Another example: let’s say you are given 6(3)^{n – 1} and you’re asked to find the first five terms. **Note**: The first term in the formula is always in the first position, you know the first term here is 6.

- First term: 6
- Second term: 6 · 3
^{2 – 1}= 6 · 3^{1}= 18 - Third term: 6 · 3
^{3 – 1}= 6 · 3^{2}= 54 - Fourth term: 6 · 3
^{4 – 1}= 6 · 3^{3}= 162 - Fifth term: 6 · 3
^{5 – 1}= 6 · 3^{4}= 486

## Sum of a Finite Geometric Sequence

The sum of a finite geometric sequence is given by the formula (Larson, 2014):

Where:

- r = common ratio (r ≠ 1),
- Σ = sigma notation (“add everything up”)

**Example question: **What is the sum of the first 7 terms of a finite geometric series if the first term (a_{1} = 1 and the common ratio (r) = 2?

## References

Harrison, B. (2020). Take Your Medicine. Adapted from Section 9.1 in Hughes-Hawlett, Deborah, et.al; Single Variable Calculus; John Wiley & Sons, Inc.; New York; 2002

Larson, R. (2014). College Algebra. Cengage Learning.

Seward, K. (2011). College Algebra: Tutorial 54D: Geometric Sequences and Series. Retrieved August 24, 2020 from: https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54d_geom.htm