C1.3 Connect nth-term formulas with function notation

Ontario Grade 11 Mathematics

Discrete Functions

Sequences as Functions: Writing, Reading, and Evaluating with f(n) and t(n)

You already know that a function takes an input, does something to it, and produces an output. In this lesson you will see that a sequence — an ordered list of numbers — behaves in exactly the same way. The position number of a term is the input, and the value of that term is the output. Connecting these two ideas lets you use the powerful language of function notation to describe any sequence with a single compact formula. This connection is the focus of expectation C1.3.

What you will learn

Recognise that a sequence is a special kind of function whose domain is the set of positive integers.
Rewrite an nth-term formula using function notation such as f(n) or t(n).
Evaluate a sequence expressed in function notation for specific values of n.
Interpret what function notation communicates about a sequence, including the input and the output.

Prerequisite Bridge: Functions and Sequences

A function is a rule that assigns exactly one output to every input. In Grade 10 you wrote functions with notation such as

f(x) = 2x + 1

, where

x

is the input and

f(x)

is the output. The expression

f(3)

means substitute

x = 3

and calculate the result:

f(3) = 2(3) + 1 = 7

.

A sequence is an ordered list of numbers, for example

3, 7, 11, 15, \ldots

Each number in the list is called a term. The first term is called

t_1

, the second is

t_2

, and so on. The subscript (small number written below) tells you the position of the term in the list.

An nth-term formula is a rule that lets you calculate any term directly from its position number

n

. For the sequence

3, 7, 11, 15, \ldots

the nth-term formula is

t_n = 4n - 1

. Substituting

n = 1

gives

4(1) - 1 = 3

, which matches the first term. Substituting

n = 4

gives

4(4) - 1 = 15

, which matches the fourth term.

A function assigns one output to every input.
A sequence is an ordered list; each term has a position number $n$ .
An nth-term formula calculates the value of any term from its position $n$ .

Why a Sequence Is a Function

Think about what a sequence really does. You give it a position number — say 5 — and it returns one specific value. That is exactly what a function does. The only difference from a function like

f(x) = 2x + 1

is that the inputs of a sequence are restricted to positive integers:

1, 2, 3, 4, \ldots

(you cannot ask for the 2.7th term of a sequence).

Because a sequence behaves like a function, mathematicians write nth-term formulas using function notation. Instead of

t_n = 4n - 1

, you can write

f(n) = 4n - 1

or

t(n) = 4n - 1

. Both forms say the same thing: input the position number

n

, and the formula outputs the value of that term.

The letter inside the brackets is always the position number. The whole expression on the left, such as

f(n)

or

t(n)

, represents the value of the term at position

n

. This is exactly the same idea as

f(x)

representing the output when the input is

x

.

t(n) = t_n

A sequence is a function whose inputs are positive integers.
The nth-term formula and the function notation form are two ways of writing the same rule.
$f(n)$ or $t(n)$ on the left means: the value of the term at position $n$ .
The domain of a sequence written as a function is $\{1, 2, 3, 4, \ldots\}$ .

Converting Between Notations and Evaluating

Converting an nth-term formula to function notation is straightforward: replace the subscript

n

in

t_n

with the bracket notation

(n)

. For example,

t_n = 3n^2 - 2

becomes

t(n) = 3n^2 - 2

. The rule itself does not change; only how you write the left side changes.

Evaluating in function notation works exactly like evaluating any function. To find

t(5)

, substitute

n = 5

into the formula and simplify. The notation

t(5)

is a compact instruction: it tells you precisely which term you need, and the formula tells you how to calculate it.

You can also reverse the process. If you are told that

f(n) = 5n + 2

and asked to find which term has value

27

, you set

f(n) = 27

and solve for

n

:

5n + 2 = 27

, so

5n = 25

, giving

n = 5

. This means the 5th term has value 27. Because

n

must be a positive integer, you should always check that your answer for

n

is a positive whole number.

t(n) = an + b

To convert $t_n$ to function notation, replace the subscript with brackets: t_n \to t(n).
To evaluate $t(k)$ , substitute $n = k$ into the formula and simplify.
To find the position of a given value, set $f(n)$ equal to that value and solve for $n$ .
Always verify that $n$ is a positive integer when working with sequences.

Reading the Notation Carefully

Function notation carries information that subscript notation does not always make obvious. When you write

t(10)

, anyone reading it immediately knows you want the 10th term of the sequence. When you write

t(n) = 6n - 4

, the whole formula is visible as one object: a function named

t

, with input variable

n

, and rule

6n - 4

.

This matters because in MCR3U you will work with both arithmetic sequences (where there is a constant difference between terms) and geometric sequences (where there is a constant ratio). Both types can be expressed in function notation. An arithmetic sequence has a linear nth-term formula, so its graph (when you plot position on the horizontal axis and term value on the vertical axis) is a set of equally spaced points lying on a straight line. A geometric sequence has an exponential nth-term formula, so its graph is a set of points lying on an exponential curve.

Remember: the graph of a sequence in function notation is always a set of discrete (separated) points, never a connected line or curve, because

n

can only be a positive integer. Connecting the points with a line would imply that inputs like

n = 1.5

are valid, which they are not for sequences.

Function notation makes the name of the sequence, the input, and the rule visible in one expression.
Arithmetic sequences produce linear nth-term formulas; geometric sequences produce exponential ones.
The graph of a sequence is always discrete (individual points), never continuous.

Putting It All Together: A Mixed Practice Perspective

At a basic level, the connection between nth-term formulas and function notation is a translation between two ways of writing the same rule. At a deeper level, it reveals that sequences are members of the larger family of functions — a key organisational idea in MCR3U.

When a problem gives you a sequence in subscript form, you can immediately rewrite it in function notation to use all the tools you know for functions: evaluating at a point, finding the input for a given output, and describing the domain. When a problem gives you a function formula and specifies that the domain is the positive integers, you know you are working with a sequence.

Always be alert to context. If a question gives you

f(n) = 2^n

and states the domain is positive integers, list the terms as

f(1) = 2

,

f(2) = 4

,

f(3) = 8

, and so on. The formula and the sequence are the same mathematical object; function notation is simply a cleaner, more flexible way to express it.

Sequences and functions are the same mathematical object when the domain is restricted to positive integers.
All function tools — evaluating, solving, describing domain — apply to sequences written in function notation.
Recognising whether a formula describes an arithmetic or geometric sequence helps you anticipate its behaviour.

Subscript Notation vs. Function Notation: Side-by-Side Comparison

Feature	Subscript Notation	Function Notation
How to write the formula	$t_n = 4n - 1$	$t(n) = 4n - 1$
How to name a specific term	$t_3$ (the 3rd term)	$t(3)$ (evaluate at $n = 3$ )
What the input is called	$n$ (subscript)	$n$ (argument in brackets)
How to calculate the 3rd term	Substitute $n = 3$ : $4(3)-1=11$	Evaluate $t(3) = 4(3)-1=11$
Domain (valid inputs)	Positive integers $n \geq 1$	Positive integers $n \geq 1$

Worked example

Converting a Subscript Formula and Evaluating Specific Terms

A sequence has nth-term formula

t_n = 5n - 3

. (a) Rewrite this formula using function notation. (b) Find

t(4)

and

t(9)

. (c) Which term of the sequence has value 42?

Rewrite in function notation
Replace the subscript $n$ with bracket notation. The name of the function stays as $t$ , and the rule $5n - 3$ does not change.
$t(n) = 5n - 3$
Evaluate at n = 4
Substitute $n = 4$ into the formula. Multiply first, then subtract, following order of operations.
$t(4) = 5(4) - 3 = 20 - 3 = 17$
Evaluate at n = 9
Substitute $n = 9$ into the formula and simplify the same way.
$t(9) = 5(9) - 3 = 45 - 3 = 42$
Set up the equation to find the position
You want to find $n$ such that $t(n) = 42$ . Set the formula equal to 42 and solve for $n$ .
$5n - 3 = 42$
Solve for n
Add 3 to both sides, then divide both sides by 5.
$5n = 45 \Rightarrow n = 9$
Verify n is a positive integer
Since $n = 9$ is a positive whole number, it is a valid position in the sequence. This confirms that $t(9) = 42$ , which agrees with part (b).

Answer: (a)

t(n) = 5n - 3

. (b)

t(4) = 17

and

t(9) = 42

. (c) The 9th term has value 42.

Check: Substitute

n = 4

:

5(4) - 3 = 17

✓. Substitute

n = 9

:

5(9) - 3 = 42

✓. Solving

5n - 3 = 42

gives

n = 9

, a positive integer ✓.

Worked example

Working Backwards from Function Notation to Describe a Sequence

A function is defined by

f(n) = 3 \cdot 2^{n-1}

where

n

is a positive integer. (a) List the first four terms of the sequence. (b) Evaluate

f(6)

. (c) Explain why

f(2.5)

is not a valid term of this sequence.

Find f(1)
Substitute $n = 1$ . The exponent becomes $1 - 1 = 0$ , and any non-zero base raised to the power 0 equals 1.
$f(1) = 3 · 2^{0} = 3 · 1 = 3$
Find f(2)
Substitute $n = 2$ . The exponent becomes $2 - 1 = 1$ .
$f(2) = 3 · 2^{1} = 3 · 2 = 6$
Find f(3) and f(4)
Continue substituting $n = 3$ and $n = 4$ to complete the list of the first four terms.
$f(3) = 3 · 2^{2} = 12, \ f(4) = 3 · 2^{3} = 24$
Evaluate f(6)
Substitute $n = 6$ . The exponent is $6 - 1 = 5$ , so calculate $2^5$ first, then multiply by 3.
$f(6) = 3 · 2^{5} = 3 · 32 = 96$
Explain why f(2.5) is invalid
The domain of this sequence is the set of positive integers $\{1, 2, 3, 4, \ldots\}$ . The value $n = 2.5$ is not a positive integer, so $f(2.5)$ does not correspond to any term of the sequence. A sequence is a function only over positive integers, so positions between whole numbers do not exist.

Answer: (a) The first four terms are 3, 6, 12, 24. (b)

f(6) = 96

. (c)

n = 2.5

is not a positive integer, so it is outside the domain of the sequence.

Check: Check the pattern: each term is double the previous one (a geometric sequence with first term 3 and common ratio 2). CAD 3, 6, 12, 24, 48, 96 — the 6th term is 96 ✓.

Common mistakes and how to avoid them

Writing

f(n)

as

f \times n

, treating the brackets as multiplication.

Correction: In function notation,

f(n)

means 'the output of function

f

when the input is

n

'. It is never multiplication. Only substitute the value of

n

into the rule.

Changing the rule when converting notation — for example, writing

t(n) = 4(n) - 1

as

t(n) = 4n

after forgetting the

-1

.

Correction: Only the left side changes when you convert from

t_n

to

t(n)

. The rule on the right side stays exactly the same.

Accepting a non-integer answer for

n

when finding which term equals a given value.

Correction: The domain of a sequence is positive integers only. If solving for

n

gives a decimal or negative result, the given value is not a term of the sequence.

Drawing a continuous line through the plotted points of a sequence instead of leaving them as separate dots.

Correction: Because

n

must be a positive integer, only discrete (individual) points exist. A connected line would suggest that inputs like

n = 1.5

are valid, but they are not.

Confusing

f(n)

with

f

times

(n)

when a number is substituted — for example, computing

f(3) = f \times 3

instead of evaluating the formula at

n = 3

.

Correction: Always replace the variable in the formula with the given number and then simplify. The function name

f

is just a label; it does not multiply anything.

Lesson summary

A sequence is a function whose domain is the set of positive integers $\{1, 2, 3, \ldots\}$ .
An nth-term formula such as $t_n = 4n - 1$ can be rewritten in function notation as $t(n) = 4n - 1$ without changing the rule.
To evaluate $t(k)$ , substitute $n = k$ into the formula and simplify.
To find which term equals a given value $v$ , set $t(n) = v$ and solve for $n$ , then verify $n$ is a positive integer.
The graph of a sequence is always a set of discrete points, never a continuous line, because the domain contains only positive integers.
Function notation makes it easier to communicate, evaluate, and analyse sequences using all the tools of functions.

Check your understanding

Question 1

Which of the following correctly rewrites

t_n = 7n + 2

in function notation?

$t(n) = 7 + 2n$
$t(n) = 7n + 2$
$f(n) = 7n \times 2$
$t_n(7) = n + 2$

Show answer and explanation

t(n) = 7n + 2

Function notation replaces the subscript

n

with bracket notation. The rule

7n + 2

stays exactly the same, giving

t(n) = 7n + 2

. The other options each change the rule incorrectly.

Question 2

Given

f(n) = 3n - 5

, what is the value of

f(7)

?

$16$
$21$
$26$
$12$

Show answer and explanation

16

Substitute

n = 7

:

f(7) = 3(7) - 5 = 21 - 5 = 16

. Option B forgets to subtract 5, option C adds incorrectly, and option D uses

3(7-5) = 6

rather than the correct order of operations.

Question 3

A sequence is defined by

t(n) = 4n + 1

. Which term of the sequence has value 29?

$n = 7$
$n = 8$
$n = 6$
$n = 28$

Show answer and explanation

n = 7

Set

4n + 1 = 29

, so

4n = 28

, giving

n = 7

. Check:

t(7) = 4(7) + 1 = 28 + 1 = 29

✓. Option B gives

t(8) = 33 \neq 29

. Option D confuses the value 28 with the position number.

Question 4

Why is

f(1.5)

not a valid term of a sequence defined by

f(n) = 2n + 3

over the positive integers?

Because $f(1.5)$ gives a negative number.
Because the domain of a sequence contains only positive integers, and 1.5 is not a positive integer.
Because $f(n) = 2n + 3$ is not a valid function.
Because 1.5 is greater than 1.

Show answer and explanation

Because the domain of a sequence contains only positive integers, and 1.5 is not a positive integer.

The domain of any sequence, when written as a function, is restricted to positive integers

\{1, 2, 3, \ldots\}

. Since 1.5 is not a positive integer, it is not a valid input. The other options are factually incorrect.

Key terms

Sequence: An ordered list of numbers where each number occupies a specific position.
Term: A single number in a sequence; the $n$ th term is the number at position $n$ .
nth-term formula: A formula that calculates the value of any term directly from its position number $n$ .
Function notation: A way of writing a function using brackets, such as $f(n)$ or $t(n)$ , to show the input and output clearly.
Domain: The set of all valid input values for a function. For a sequence, the domain is the positive integers.
Evaluate: To substitute a specific value for the variable in a formula and calculate the result.
Discrete: Consisting of separate, individual points with no values in between; the opposite of continuous.
Positive integers: The counting numbers $1, 2, 3, 4, \ldots$ ; the valid position numbers for terms in a sequence.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation C1.3. It is a study resource, not an official curriculum publication.

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C1.3 · Connect nth-term formulas with function notation

Ontario Grade 11 Mathematics

Sequences as Functions: Writing, Reading, and Evaluating with f(n) and t(n)

What you will learn

Prerequisite Bridge: Functions and Sequences

Why a Sequence Is a Function

Converting Between Notations and Evaluating

Reading the Notation Carefully

Putting It All Together: A Mixed Practice Perspective

Subscript Notation vs. Function Notation: Side-by-Side Comparison

Converting a Subscript Formula and Evaluating Specific Terms

Working Backwards from Function Notation to Describe a Sequence

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCR3U

About this lesson