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Geometric Sequences & Exponential Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 10 April 2026

Geometric sequences & exponential functions

What is an exponential function?

An exponential function is a function of the form $f (x) = a b^{x}$
- where $a$ is the initial value (the output when $x = 0$ )
- and $b$ is the base

Key features:
- The base $b$ must satisfy $b > 0$ and $b \neq 1$
  - if $b = 1$ , then $f (x) = a$ , which is just a constant function
    - i.e. a linear function with a horizontal graph
- The domain of an exponential function is all real numbers
- If $a > 0$ and $b > 1$ , the function demonstrates exponential growth
- If $a > 0$ and $0 < b < 1$ , the function demonstrates exponential decay

Examiner Tips and Tricks

You should already have basic familiarity with exponential functions from your study prior to AP® Precalculus.

A geometric sequence of the form $g_{n} = g_{0} r^{n}$ and an exponential function of the form $f (x) = a b^{x}$ have the same structure
Both can be expressed as
- an initial value
- with repeated multiplication by a constant ratio

	Geometric sequence	Exponential function
Formula	$g_{n} = g_{0} r^{n}$	$f (x) = a b^{x}$
Initial value	$g_{0}$	$a$
Constant ratio	$r$ (common ratio)	$b$ (base)

In both cases
- each time the input increases by 1
- the output is multiplied by the same constant ( $r$ or $b$ )
This means that the terms of a geometric sequence
- are the same as the values of a corresponding exponential function evaluated at whole number inputs
- E.g. the geometric sequence $g_{n} = 5 \cdot 2^{n}$ gives the same output values as the exponential function $f (x) = 5 \cdot 2^{x}$
  - but only at $x = 0, 1, 2, 3, \dots$

How does the known-point form of an exponential function relate to the geometric sequence formula?

The geometric sequence formula based on a known term, $g_{n} = g_{k} r^{(n - k)}$
- has a direct parallel in the known-point form of an exponential function
An exponential function can be written in known-point form as $f (x) = y_{i} r^{(x - x_{i})}$
- where $(x_{i}, y_{i})$ is a known point on the graph
- and $r$ is a known ratio
Compare these side by side:

	Geometric sequence	Exponential function
Formula	$g_{n} = g_{k} r^{(n - k)}$	$f (x) = y_{i} r^{(x - x_{i})}$
Known value	$g_{k}$ (the $k$ th term)	$y_{i}$ (the output at $x_{i}$ )
Constant ratio	$r$	$r$
Difference from known input	$n - k$	$x - x_{i}$

Both say the same thing:
- Start at a known output value
- then multiply by the constant ratio raised to the power of how far the input is from the known input
This means you can construct an exponential function from two data points
- using the same logic as finding the general term of a geometric sequence from two terms

Examiner Tips and Tricks

When constructing an exponential model from two data points, use the known-point form

find the ratio $r$ from the ratio of the two output values
then write $f (x) = y_{i} r^{(x - x_{i})}$

How do the domains of an exponential function and a geometric sequence differ?

Although a geometric sequence and an exponential function can share the same formula
- they have different domains
A geometric sequence is defined only for whole number inputs ( $n = 0, 1, 2, 3, \dots$ )
- The graph of a geometric sequence is a set of discrete points
An exponential function is defined for all real numbers
- The graph of the corresponding exponential function is a smooth continuous curve
  - that passes through all of those points
  - and extends between and beyond them

Graph showing exponential growth of f(x) = 0.3 * 1.6^x with points marked on curve, x-axis as x or n, y-axis as y or a_n, labelled axes. — **Graphs of an exponential function and the corresponding geometric sequence on the same set of axes**

In many contextual problems, the geometric sequence captures values at discrete time steps (e.g. after each year)
- while the exponential function extends the model to allow predictions at any input value (e.g. after 2.5 years)

Examiner Tips and Tricks

The exam frequently presents tables of data and asks you to determine the function type.

If you check the ratios between successive output values over equal-length input intervals and find they are constant
- then the data is exponential
This is the same as recognizing a geometric pattern

Worked Example

A colony of bacteria is being studied. The number of bacteria, in thousands, is recorded at regular intervals. The table below shows the data.

Time (hours)	0	1	2	3	4
Bacteria (thousands)	8	12	18	27	40.5

(a) Show that the bacteria data can be modeled by an exponential function.

Answer:

Check the ratios between successive output values over equal-length input intervals

Each interval in the table has length 1 hour, so the intervals are all equal-length

$\frac{12}{8} = 1.5$

$\frac{18}{12} = 1.5$

$\frac{27}{18} = 1.5$

$\frac{40.5}{27} = 1.5$

The ratios are constant (all equal to 1.5), so the output values change proportionally. This means the data can be modeled by an exponential function.

(b) Write an exponential function $B (t)$ that models the number of bacteria, in thousands, as a function of time $t$ , in hours.

Answer:

Use $f (x) = a b^{x}$

The constant ratio (base) is $b = 1.5$
The initial value (at $t = 0$ ) is $a = 8$

$B (t) = 8 \cdot 1 . 5^{t}$

Answer:

Substitute $t = 2.5$ into the equation from part (b)

$B (2.5) = 8 \cdot 1 . 5^{2.5} = 8 \cdot 2.756 = 22.045407 . . .$

Round to 3 decimal places

The estimated number of bacteria at $t = 2.5$ hours is approximately 22.045 thousand (or about 22,045 bacteria)

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Geometric Sequences & Exponential Functions (College Board AP® Precalculus): Study Guide

Geometric sequences & exponential functions

What is an exponential function?

Examiner Tips and Tricks

How does the known-point form of an exponential function relate to the geometric sequence formula?

Examiner Tips and Tricks

How do the domains of an exponential function and a geometric sequence differ?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis

Geometric Sequences & Exponential Functions (College Board AP® Precalculus): Study Guide

Geometric sequences & exponential functions

What is an exponential function?

How are geometric sequences related to exponential functions?

How does the known-point form of an exponential function relate to the geometric sequence formula?

How do the domains of an exponential function and a geometric sequence differ?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis