Pre-Algebra
Number sequences
Geometric sequences
20 practice questions
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Theory + worked examples
Geometric Sequences
California Pre-Algebra • Standard F-BF.2 • Number Sequences
Geometric Sequences is a topic in Number Sequences in the California Common Core State Standards. It is aligned to Standard F-BF.2, which requires students to identify and extend geometric sequences and find a general term.
A geometric sequence multiplies by a constant ratio between terms; its \(n\)th term is \(a_1 r^{\,n-1}\).
Theory
A geometric sequence multiplies by a constant common ratio \(r\) from term to term.
The \(n\)th term is \(a_n=a_1\cdot r^{\,n-1}\).
Terms grow by a constant factor.
A geometric sequence.
The nth term:
\[a_n=a_1\cdot r^{\,n-1}\]
\(r\) is the constant ratio between terms.
How to work with a geometric sequence
- Divide consecutive terms to find \(r\).
- Multiply by \(r\) to get the next term.
- Use \(a_n=a_1 r^{n-1}\) for any term.
- A ratio between \(0\) and \(1\) means decay.
Example 1 — Common ratio
Find \(r\) for \(2,6,18,54,\ldots\)
Solution
Divide consecutive terms.
| \(\dfrac{6}{2}\) | \(=\) | \(3\) |
Example 2 — Next term
Find the next term of \(2,6,18,54,\ldots\)
Solution
Multiply by \(3\).
| \(54\cdot3\) | \(=\) | \(162\) |
Example 3 — nth term
Find the \(5\)th term of \(2,6,18,\ldots\)
Solution
Use \(a_1 r^{n-1}\).
| \(2\cdot3^{4}\) | \(=\) | \(162\) |
Example 4 — Decay
Find \(r\) for \(16,8,4,2,\ldots\)
Solution
Divide.
| \(\dfrac{8}{16}\) | \(=\) | \(\dfrac{1}{2}\) |
Common pitfalls
Geometric multiplies; arithmetic adds.
The exponent is \(n-1\), not \(n\).
Find \(r\) by dividing, not subtracting.
Frequently asked questions
What is a geometric sequence?
One that multiplies by a constant ratio.
What is the common ratio?
The constant factor between terms.
Next term of \(2,6,18,54\)?
\(162\).
What is the nth-term formula?
\(a_n=a_1 r^{n-1}\).
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