Geometric Sequences
Each term is the previous times a constant ratio.
About this topic
In a geometric sequence each term is found by multiplying the previous term by a constant common ratio r. For 2, 6, 18, 54, … r = 3, and the nth term is aₙ = a₁ · rⁿ⁻¹. If the absolute value of r is less than 1 the terms shrink, and if r is negative the signs alternate.
Geometric sequences are exponential functions with whole-number inputs. The ratio is found by dividing any term by the one before it. To get the 6th term of 2, 6, 18, …, compute 2 · 3⁵ = 486. Notice how quickly the numbers explode compared with an arithmetic sequence that adds 4 each time.
They describe repeated percentage change: a bouncing ball that returns to 60 percent of its previous height, a population doubling each hour, a car losing 15 percent of its value yearly, or money earning compound interest. Viral spread of a rumor or video often starts out geometric.
A typical error is to apply the exponent n instead of n − 1, which shifts every term. Another is to mistake a ratio for a difference; always divide to test. Remember that r = 1 gives a constant sequence and r = 0 is not useful.
Key ideas
- aₙ = a₁ rⁿ⁻¹
- r is the common ratio
- Growth is exponential
Quick check
Use cases: where Geometric Sequences helps
- Homework: review the key idea behind Geometric Sequences before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Geometric Sequences without notes.
- Visual learning: play with the interactive lab to see how Geometric Sequences changes when you adjust its inputs.
- Teaching a friend: explain Geometric Sequences in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Geometric Sequences?
Each term is the previous times a constant ratio.
What will I learn in Geometric Sequences?
aₙ = a₁ rⁿ⁻¹. r is the common ratio. Growth is exponential.
Is Geometric Sequences part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Geometric Sequences lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Geometric Sequences?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.