Algebra 2
Sequences and series
Geometric sequences and series
20 practice questions
0 video lessons
Theory + worked examples
Geometric Sequences and Series
Common Core Algebra 2 • Standard A-SSE.4 • Sequences & Series
Geometric Sequences and Series is a topic in Sequences & Series in the Common Core State Standards. It is aligned to Standard A-SSE.4, which requires students to derive and use the formula for the sum of a finite geometric series.
A geometric sequence multiplies by a constant ratio; its nth term is \(a_1 r^{n-1}\) and the sum is \(\dfrac{a_1(1-r^n)}{1-r}\).
Theory
A geometric sequence multiplies by a constant common ratio \(r\):
\[a_n=a_1\,r^{\,n-1},\qquad S_n=\dfrac{a_1(1-r^n)}{1-r}\ (r\neq1).\]
The terms grow (or shrink) exponentially — each is a fixed multiple of the last.
Terms multiply by \(r=2\) each step.
The geometric formulas.
nth term and sum:
\[a_n=a_1\,r^{\,n-1},\qquad S_n=\dfrac{a_1(1-r^n)}{1-r}\]
The exponent is \(n-1\) for the \(n\)th term.
How to work with geometric sequences
- Find \(a_1\) and the common ratio \(r\).
- Use \(a_n=a_1 r^{n-1}\) for a term.
- Use \(S_n=\dfrac{a_1(1-r^n)}{1-r}\) for a sum.
- Divide consecutive terms to find \(r\).
Example 1 — nth term
Find the \(5\)th term of \(2,6,18,54,\dots\).
Solution
Here \(a_1=2,\ r=3\).
| \(a_5\) | \(=\) | \(2\cdot3^{4}\) |
| \(=\) | \(2(81)=162\) |
Example 2 — Sum of terms
Find the sum of the first \(5\) terms of \(2,6,18,\dots\).
Solution
Use \(S_n=\dfrac{a_1(1-r^n)}{1-r}\).
| \(S_5\) | \(=\) | \(\dfrac{2(1-3^5)}{1-3}\) |
| \(=\) | \(\dfrac{2(-242)}{-2}=242\) |
Example 3 — Find the ratio
A geometric sequence has \(a_1=3\) and \(a_3=12\). Find \(r\) (positive).
Solution
Use \(a_3=a_1r^2\).
| \(12\) | \(=\) | \(3r^2\) |
| \(r\) | \(=\) | \(2\) |
Example 4 — Growth pattern
A culture doubles every hour from \(100\). Find the count after \(4\) hours.
Solution
Geometric with \(r=2\); after 4 hours use the 5th term.
| \(a\) | \(=\) | \(100\cdot2^{4}\) |
| \(=\) | \(1600\) |
Common pitfalls
The exponent is \(n-1\), not \(n\).
A common ratio is multiplied, not added.
\(r\) can be negative — terms then alternate sign.
Frequently asked questions
What is a geometric sequence?
A sequence with a constant ratio between terms.
What is the nth term formula?
\(a_n=a_1 r^{n-1}\).
How do you find the common ratio?
Divide any term by the one before it.
How do you sum a geometric series?
\(S_n=\dfrac{a_1(1-r^n)}{1-r}\).
More in Sequences and series