Geometric Sequence
In short: A geometric sequence is a list in which every entry equals the entry before it times one steady multiplier, the common ratio.
Algebra 1Functions and GraphsF-IF.A.3
What is Geometric Sequence?
Each rebound of a dropped ball falls short of the last by the same fraction, and those heights form a geometric sequence: instead of adding a constant from entry to entry, you multiply by one.
To find the common ratio \(r\), divide any entry by its predecessor. When r is greater than 1 the entries grow fast, when r is between 0 and 1 they shrink toward zero, and when \(r\) is negative the signs alternate. The \(n\)th entry is \(a_n = a_1 \cdot r^{n-1}\), which is an exponential pattern in n. A ball bouncing back to two thirds of its previous height each time, a social post whose audience triples daily, and a loan balance growing by a fixed percent all follow this multiplying rule.
How to spot it
- Terms double, triple or shrink by half, a third or a percent at each step.
- A situation repeats a percent increase or decrease, such as a bounce or a growing audience.
- Dividing neighboring terms gives the same number every time.
- A graph of the terms curves upward or flattens toward the axis instead of forming a straight line.
The method
- Divide neighboring entries and confirm that the quotient r never changes.
- Pick out the opening entry \(a_1\).
- Place both into \(a_n = a_1 \cdot r^{n-1}\), using the position n you want.
- Work out the power before multiplying by \(a_1\).
Geometric Sequence: solved example
Problem
A rubber ball’s first rebound reaches 54 inches, and every rebound is \(\frac{2}{3}\) as high as the one before. How high is the 4th rebound?
Solution
This is a geometric sequence with \(a_1 = 54\) and \(r = \frac{2}{3}\). Use \(a_n = a_1 r^{n-1}\) with \(n = 4\).
\(a_4 = 54 \cdot \left(\frac{2}{3}\right)^3 = 54 \cdot \frac{8}{27} = 16\). The 4th rebound is 16 inches high.
Try it
Question 1
Find the 6th term of \(5, 15, 45, \ldots\).
Show the answer
\(r = 3\), so \(a_6 = 5 \cdot 3^5 = 5 \cdot 243 = 1{,}215\).
Question 2
What are the next two terms of \(2, -6, 18, -54, \ldots\)?
Show the answer
\(r = -3\), so the next terms are \(162\) and \(-486\).
Geometric Sequence vs. Arithmetic Sequence
Here the step is a multiplier, so the dots of the graph bend like exponential growth or decay. In an arithmetic list the step is an addition, so the dots stay on one straight line. Subtraction of neighbors tests the second; division of neighbors tests this one.
Keep going
Lessons
Worksheets and quizzes
Related words
- Arithmetic Sequence
- Explicit Formula
- Recursive Formula
- Exponential Function
- Geometric Series
- Exponential Growth
Whole dictionary: A–Z · Algebra 1 · Functions and Graphs