Geometric Sequence Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Geometric Sequence.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio r.
Multiply by the same number each step — 2, 6, 18, 54, ... (multiply by 3). This is exponential growth.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Geometric sequences represent exponential growth (|r|>1) or decay (|r|<1) — their graph is an exponential curve.
Common stuck point: If |r| < 1, terms shrink toward zero. If |r| > 1, terms grow without bound.
Sense of Study hint: Divide any term by the previous one to find r, then check that ratio stays constant throughout.
Worked Examples
Example 1
easySolution
- 1 Use the geometric sequence formula a_n = a_1 \cdot r^{n-1} with a_1 = 5, r = 2, n = 8.
- 2 Find the 8th term: a_8 = 5 \cdot 2^7 = 5 \cdot 128 = 640
- 3 Apply the partial sum formula: S_8 = a_1 \cdot \frac{1-r^8}{1-r} = 5 \cdot \frac{1-256}{1-2} = 5 \cdot 255 = 1275
Answer
Example 2
mediumExample 3
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
hardRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.