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Arithmetic Sequences & Linear Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 10 April 2026

Arithmetic sequences & linear functions

An arithmetic sequence of the form $a_{n} = a_{0} + d n$
- and a linear function of the form $f (x) = b + m x$
- have the same structure
Both can be expressed as
- an initial value
- plus repeated addition of a constant rate of change

	Arithmetic sequence	Linear function
Formula	$a_{n} = a_{0} + d n$	$f (x) = b + m x$
Initial value	$a_{0}$	$b$
Constant rate of change	$d$ (common difference)	$m$ (slope)

In both cases,
- each time the input increases by 1
- the output increases by the same constant amount ( $d$ or $m$ )
This means that the terms of an arithmetic sequence
- are the same as the values of a corresponding linear function evaluated at whole number inputs
- E.g. the arithmetic sequence $a_{n} = 5 + 3 n$ gives the same output values as the linear function $f (x) = 5 + 3 x$
  - but only at $x = 0, 1, 2, 3, \dots$

How does the point-slope form of a linear function relate to the arithmetic sequence formula?

The arithmetic sequence formula based on a known term, $a_{n} = a_{k} + d (n - k)$
- has a direct parallel in the point-slope form of a linear function
A linear function can be written in point-slope form as $f (x) = y_{i} + m (x - x_{i})$
- where $(x_{i}, y_{i})$ is a known point on the line
- and $m$ is the slope
Compare these side by side:

	Arithmetic sequence	Linear function
Formula	$a_{n} = a_{k} + d (n - k)$	$f (x) = y_{i} + m (x - x_{i})$
Known value	$a_{k}$ (the $k$ th term)	$y_{i}$ (the output at $x_{i}$ )
Rate of change	$d$	$m$
Difference from known input	$n - k$	$x - x_{i}$

Both say the same thing:
- Start at a known output value
- then add the rate of change multiplied by how far the input is from the known input
This means you can construct a linear function from two data points
- using the same logic as finding the general term of an arithmetic sequence from two terms

Examiner Tips and Tricks

When constructing a linear model from two data points, use point-slope form:

find the slope $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
then write $f (x) = y_{1} + m (x - x_{1})$

This works whether the inputs are whole numbers or not.

How do the domains of a linear function and linear sequence differ?

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

Graph showing a linear function y = f(x) = 3 + 2x, with points at integer x-values representing the values of the arithmetic sequence 3+2n. Axes are labelled y (or aₙ) and x (or n). — **Graphs of a linear function and the corresponding arithmetic sequence on the same set of axes**

In many contextual problems, the arithmetic sequence captures values at discrete time steps (e.g. year 1, year 2, year 3)
- while the linear function extends the model to allow predictions at any input value (e.g. year 1.5)

Examiner Tips and Tricks

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

If you check the differences between successive output values over equal-length input intervals and find they are constant
- then the data is linear
This is the same as recognizing an arithmetic pattern

Worked Example

A water tank is being filled with water. The table below shows the volume of water in the tank, in gallons, at selected times.

Time (hours)	0	1	2	3	4
Volume (gallons)	150	210	270	330	390

(a) Show that the volume data can be modeled by a linear function.

Answer:

Check the differences 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

$210 - 150 = 60$

$270 - 210 = 60$

$330 - 270 = 60$

$390 - 330 = 60$

The differences are constant (all equal to 60), so the output values change at a constant rate. This means the data can be modeled by a linear function.

(b) Write a linear function $V (t)$ that models the volume of water in the tank, in gallons, as a function of time $t$ , in hours.

Answer:

You can use the initial value form $f (x) = b + m x$

The initial value (at $t = 0$ ) is $b = 150$
The constant rate of change (slope) is $m = \frac{60}{1} = 60$ gallons per hour

$V (t) = 150 + 60 t$

Answer:

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

$\begin{array}{rcl} V (2.5) & = & 150 + 60 (2.5) \\ = & 150 + 150 \\ = & 300 \end{array}$

The estimated volume of water in the tank at $t = 2.5$ hours is 300 gallons

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Arithmetic Sequences & Linear Functions (College Board AP® Precalculus): Revision Note

Arithmetic sequences & linear functions

How does the point-slope form of a linear function relate to the arithmetic sequence formula?

Examiner Tips and Tricks

How do the domains of a linear function and linear 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

Arithmetic Sequences & Linear Functions (College Board AP® Precalculus): Revision Note

Arithmetic sequences & linear functions

How are arithmetic sequences related to linear functions?

How does the point-slope form of a linear function relate to the arithmetic sequence formula?

How do the domains of a linear function and linear sequence differ?

Unlock more, it's free!

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

Author: Roger B

Reviewer: Mark Curtis