Geometric Series
Sum of a geometric sequence.
Key idea
Sum of a geometric sequence.
In depth
A geometric series sums terms with constant ratio r: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Multiply the sum by r and subtract to see why most terms cancel.
Worked example
2 + 6 + 18 + 54 = 2(1 − 3⁴)/(1 − 3) = 2(−80)/(−2) = 80.
Formula
Real-world uses
- Calculating total payments on a loan with interest
- The total of a doubling reward in a story
Common mistake
Using the formula with r = 1, where it is undefined; the sum is just n·a.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Geometric Series?
Sum of a geometric sequence. It is a topic at the advanced level in Algebra.
What is the formula for Geometric Series?
The key formula is Sₙ = a(1−rⁿ)/(1−r).
Where is Geometric Series used in real life?
Real-world examples: Calculating total payments on a loan with interest; The total of a doubling reward in a story.
What should I learn before Geometric Series?
Make sure you are comfortable with Arithmetic Series, the previous topic in Algebra.
What is a common mistake with Geometric Series?
Using the formula with r = 1, where it is undefined; the sum is just n·a.
How can I practise Geometric Series?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Geometric Series?
Move on to Infinite Geometric Series next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.