How to Solve Geometric Sequences? (+FREE Worksheet!)
Geometric sequences are number patterns where each term is found by multiplying the previous term by the same fixed number, called the common ratio. They appear in exponential growth and decay, financial calculations, and many real-world models. Understanding geometric sequences gives you a powerful tool for analyzing patterns and predicting future values in Algebra 1 and beyond.
What Is a Geometric Sequence?
A geometric sequence is a list of numbers in which each term after the first is obtained by multiplying the previous term by a constant called the common ratio (\(\color{blue}{r}\)). If \(\color{blue}{r > 1}\), the sequence grows; if \(\color{blue}{0 < r < 1}\), the sequence shrinks; if \(\color{blue}{r < 0}\), the terms alternate in sign.
Example: \(\color{blue}{3, 6, 12, 24, 48, \ldots}\) The common ratio is \(\color{blue}{r = 2}\).
Key Formulas for Geometric Sequences
Finding the Common Ratio
Divide any term by the term before it:
\(\color{blue}{r = \frac{a_{n}}{a_{n-1}}}\)
Example: in \(\color{blue}{81, 27, 9, 3, \ldots}\) the ratio is \(\color{blue}{r = \frac{27}{81} = \frac{1}{3}}\).
Explicit (nth-Term) Formula
The nth term of a geometric sequence is:
an = a1 · r\(\color{blue}{n-1}\)
where \(\color{blue}{a_{1}}\) is the first term and \(\color{blue}{r}\) is the common ratio.
Recursive Formula
Each term is the previous term multiplied by \(\color{blue}{r}\):
\(\color{blue}{a_{n} = a_{n-1} \cdot r}\), with \(\color{blue}{a_{1}}\) given.
Step-by-Step Summary
- Identify the first term \(\color{blue}{a_{1}}\).
- Find the common ratio: \(\color{blue}{r = \frac{a_{2}}{a_{1}}}\) (verify with other consecutive pairs).
- Write the explicit formula: \(\color{blue}{a_{n} = a_{1} \cdot r^{n-1}}\).
- Substitute \(\color{blue}{n}\) to find any desired term.
Watch: Introduction to Geometric Sequences (Video Lesson)
Khan Academy introduces geometric sequences, common ratios, and the intuition behind the formula:
Geometric Sequences – Worked Examples
Example 1: Find the 5th term of the sequence \(\color{blue}{3, 6, 12, 24, \ldots}\)
\(\color{blue}{a_{1} = 3}\), \(\color{blue}{r = 2}\).
\(\color{blue}{a_{5} = 3 \cdot 2^{4} = 3 \cdot 16 = 48}\)
Example 2: Find the 5th term of \(\color{blue}{81, 27, 9, 3, \ldots}\)
\(\color{blue}{a_{1} = 81}\), \(\color{blue}{r = \frac{1}{3}}\).
\(\color{blue}{a_{5} = 81 \cdot (\frac{1}{3})^{4} = 81 \cdot \frac{1}{81} = 1}\)
Example 3: Find the 5th term of \(\color{blue}{2, -6, 18, -54, \ldots}\)
\(\color{blue}{a_{1} = 2}\), \(\color{blue}{r = -3}\).
\(\color{blue}{a_{5} = 2 \cdot (-3)^{4} = 2 \cdot 81 = 162}\)
Example 4: Find the common ratio when \(\color{blue}{a_{1} = 5}\) and \(\color{blue}{a_{4} = 40}\).
\(\color{blue}{40 = 5 \cdot r^{3} \Rightarrow r^{3} = 8 \Rightarrow r = 2}\)
More Practice: Using Explicit Formulas of Geometric Sequences (Video)
Khan Academy demonstrates how to apply the explicit formula to find specific terms and work backwards to find the common ratio:
Exercises for Geometric Sequences
- Find the common ratio: \(\color{blue}{5, 15, 45, 135, \ldots}\)
- Find the 6th term: \(\color{blue}{2, 6, 18, 54, \ldots}\)
- Find the 4th term: \(\color{blue}{256, 64, 16, \ldots}\)
- Write the explicit formula for: \(\color{blue}{4, 8, 16, 32, \ldots}\)
- Find \(\color{blue}{a_{7}}\) when \(\color{blue}{a_{1} = 1}\) and \(\color{blue}{r = -2}\).
Answers
- \(\color{blue}{r = 3}\)
- \(\color{blue}{a_{6} = 2 \cdot 3^{5} = 486}\)
- \(\color{blue}{a_{4} = 256 \cdot (\frac{1}{4})^{3} = 4}\)
- \(\color{blue}{a_{n} = 4 \cdot 2^{n-1}}\)
- \(\color{blue}{a_{7} = 1 \cdot (-2)^{6} = 64}\)
Frequently Asked Questions
What is the difference between a geometric and an arithmetic sequence?
In an arithmetic sequence, you add the same number (common difference) to get the next term. In a geometric sequence, you multiply by the same number (common ratio). Example: \(\color{blue}{2, 5, 8, 11}\) is arithmetic (add 3); \(\color{blue}{2, 6, 18, 54}\) is geometric (multiply by 3).
Can the common ratio be negative?
Yes. A negative common ratio causes the terms to alternate between positive and negative. For example, \(\color{blue}{r = -2}\) with \(\color{blue}{a_{1} = 1}\) gives \(\color{blue}{1, -2, 4, -8, 16, \ldots}\)
What if the common ratio is between 0 and 1?
The sequence decreases toward zero. For example, \(\color{blue}{r = \frac{1}{2}}\) with \(\color{blue}{a_{1} = 64}\) gives \(\color{blue}{64, 32, 16, 8, 4, 2, 1, \ldots}\) — each term is half the previous one.
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