Connect sequences with discrete functions

Ontario Grade 11 Mathematics

Discrete Functions

Connecting Patterns to Function Notation (MCR3U – C1.1)

You have worked with functions before — rules that pair every input with exactly one output. In this lesson you will see that a sequence (an ordered list of numbers that follows a pattern) is actually a special kind of function. The key difference from the functions you saw in Grade 10 is that the inputs are restricted to counting numbers: 1, 2, 3, 4, and so on. That restriction turns out to be very useful for modelling real situations where things happen in distinct steps, like monthly savings deposits or the number of seats in expanding rows of a theatre. By the end of this lesson you will be able to move fluently between a sequence written as a list and the same sequence written in function notation.

What you will learn

Explain what makes a sequence a discrete function by identifying its domain, range, and rule.
Write the terms of a sequence using both list notation and function notation.
Determine whether a given sequence is arithmetic or geometric by examining how consecutive terms relate.
Evaluate a sequence function at specific natural-number inputs and interpret what the output means.

Prerequisite Bridge: What Is a Function?

Recall from Grade 10 that a function is a rule that assigns exactly one output value to each input value. You can think of it as a machine: drop in an input, get out one specific output. We write this as

f(x)

, read 'f of x', where

x

is the input.

A function also has a domain (the set of all allowed inputs) and a range (the set of all resulting outputs). For example, the linear function

f(x) = 3x + 1

has domain 'all real numbers' and produces any real number as output.

The functions in this lesson use a much smaller, very specific domain: the natural numbers

\{1, 2, 3, 4, \dots\}

. This restriction is what makes a sequence a discrete function — 'discrete' means the inputs are separated, with gaps between them, rather than forming a continuous line.

A function pairs each input with exactly one output.
Domain = set of allowed inputs; range = set of resulting outputs.
Discrete means the domain contains only separated values, not every point on a number line.
In a sequence, the domain is the set of positive integers (term numbers).

What Is a Sequence?

A sequence is an ordered list of numbers, called terms, arranged according to a pattern or rule. We label each term by its position: the first term is

t_1

, the second is

t_2

, the third is

t_3

, and so on. The subscript (the small number below the

t

) is the term number, or index.

Consider the sequence

2, 5, 8, 11, 14, \dots

Each term is 3 more than the one before it. This constant addition of 3 is the pattern. If someone asks for the 6th term, you can find it because the rule is consistent.

Because each term number (1, 2, 3, …) maps to exactly one term value, a sequence satisfies the definition of a function. The term number is the input; the term value is the output. This is why we call a sequence a discrete function.

t_n = f(n), n ∈ \{1, 2, 3, \dots\}

A sequence is an ordered list of numbers following a consistent rule.
Each position in the list is called a term number or index.
A sequence is discrete because term numbers are positive integers, not all real numbers.
A sequence qualifies as a function because each term number gives exactly one term value.

Function Notation for Sequences

Because a sequence is a discrete function, we can write it using function notation. Instead of writing a list, we write a formula where

n

is the input (term number) and

f(n)

is the output (term value). The domain is stated explicitly as the positive integers.

Take the sequence

2, 5, 8, 11, \dots

again. Each term equals

3n - 1

when

n

is its position. Check:

f(1) = 3(1) - 1 = 2

✓,

f(2) = 3(2) - 1 = 5

✓,

f(3) = 3(3) - 1 = 8

✓. The formula works for every term.

This sequence is called arithmetic because consecutive terms differ by the same constant, called the common difference

d

. A general arithmetic sequence can always be written as

f(n) = a + (n-1)d

, where

a

is the first term and

d

is the common difference. Expanding this gives a linear function of

n

, which explains why arithmetic sequences, when graphed, produce points that lie on a straight line.

A geometric sequence, by contrast, multiplies by the same constant ratio

r

each time. Its function rule is

f(n) = a \cdot r^{n-1}

, where

a

is the first term. Because the variable

n

is in the exponent, this is an exponential function of

n

, not a linear one.

f(n) = a + (n-1)d or f(n) = a · r^{n-1}

Sequence function notation: $f(n)$ gives the value of the $n$ -th term.
An arithmetic sequence has a constant common difference $d$ and is a linear function of $n$ .
A geometric sequence has a constant common ratio $r$ and is an exponential function of $n$ .
The domain of any sequence function is always restricted to positive integers.

The Graph of a Sequence: Discrete vs. Continuous

When you graphed

f(x) = 3x - 1

in Grade 10, you drew a solid line because every real-number value of

x

was allowed. A sequence version of the same rule,

f(n) = 3n - 1

with n ∈ \{1, 2, 3, \dots\}, looks completely different on a graph: you get isolated dots at

(1, 2)

,

(2, 5)

,

(3, 8)

, and so on, with nothing drawn between them.

This distinction is critical. You must never connect the dots of a sequence graph with a line or curve, because non-integer inputs (like

n = 1.5

) have no meaning in the sequence — there is no 'term 1.5'. The dots are deliberately isolated to show that the function is discrete.

The same idea applies to geometric sequences. The sequence

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

gives the points

(1, 3)

,

(2, 6)

,

(3, 12)

,

(4, 24)

. These points curve upward, just like an exponential graph, but they are plotted as separate dots, not as a smooth curve.

Sequence graphs show isolated dots, never a connected line or curve.
Only positive integer values of $n$ are plotted.
Arithmetic sequence graphs have dots that would lie on a line if connected.
Geometric sequence graphs have dots that would lie on an exponential curve if connected.

Recognising and Using the Connection

To confirm that a sequence is a discrete function, check two things: (1) every term number in the domain produces exactly one term value, and (2) the domain is a set of positive integers. If both are true, you can write the sequence as

f(n)

with n ∈

\mathbb{N}

.

Recognising the type of sequence helps you choose the right formula. If consecutive differences are constant, use the arithmetic formula. If consecutive ratios are constant, use the geometric formula. To find the ratio between consecutive terms, divide any term by the one before it:

r = \frac{t_{n}}{t_{n-1}}

.

Once you have the formula, function notation lets you answer questions precisely. 'What is the 20th term?' becomes 'Evaluate

f(20)

.' 'Which term equals 98?' becomes 'Solve

f(n) = 98

for

n

, where

n

must be a positive integer.' This connection between sequences and discrete functions gives you a powerful, organised way to work with patterns.

r = \frac{t_n}{t_{n-1}}

Verify a sequence is a discrete function by confirming each term number gives one output and the domain is positive integers.
Use constant differences to identify arithmetic sequences; use constant ratios for geometric sequences.
Function notation turns word questions about sequences into precise mathematical statements.
Solving $f(n) = k$ finds which term has value $k$ , and $n$ must be a positive integer for the answer to be valid.

Arithmetic vs. Geometric Sequences as Discrete Functions

Feature	Arithmetic Sequence	Geometric Sequence
Pattern rule	Add a constant $d$ each time	Multiply by a constant $r$ each time
Function formula	$f(n) = a + (n-1)d$	$f(n) = a \cdot r^{n-1}$
Type of function	Linear in $n$	Exponential in $n$
Graph appearance	Dots on a straight line	Dots on a curve
Domain	Positive integers $\{1, 2, 3, \dots\}$	Positive integers $\{1, 2, 3, \dots\}$

Worked example

Writing an Arithmetic Sequence as a Discrete Function

A sequence begins:

7, 11, 15, 19, \dots

(a) Identify the type of sequence and find the common difference. (b) Write the sequence as a discrete function

f(n)

. (c) Find

f(12)

and explain what it represents. (d) Determine which term of the sequence has the value

63

.

Identify the type and common difference
Subtract each term from the next: $11 - 7 = 4$ , $15 - 11 = 4$ , $19 - 15 = 4$ . The difference is constant at $4$ , so this is an arithmetic sequence with first term $a = 7$ and common difference $d = 4$ .
$d = 4, a = 7$
Write the general function rule
Use the arithmetic sequence formula $f(n) = a + (n-1)d$ . Substitute $a = 7$ and $d = 4$ , then simplify by expanding and collecting like terms.
$f(n) = 7 + (n-1)(4)$
Simplify the formula
Expand the bracket: $(n-1)(4) = 4n - 4$ . Then add $7$ : $7 + 4n - 4 = 4n + 3$ . The simplified rule is $f(n) = 4n + 3$ , valid for n ∈ \{1, 2, 3, \dots\}.
$f(n) = 4n + 3, n ∈ \{1, 2, 3, \dots\}$
Evaluate f(12)
Substitute $n = 12$ into the formula. Multiply first, then add, following order of operations.
$f(12) = 4(12) + 3 = 48 + 3 = 51$
Interpret f(12)
Because $f(12) = 51$ , the 12th term of the sequence is $51$ . In other words, when the term number (input) is $12$ , the term value (output) is $51$ .
Find which term equals 63
Set $f(n) = 63$ and solve for $n$ . Subtract $3$ from both sides, then divide by $4$ . Check that the result is a positive integer so it is a valid term number.
$4n + 3 = 63 \implies 4n = 60 \implies n = 15$

Answer: The sequence is arithmetic. The function is

f(n) = 4n + 3

, n ∈ \{1, 2, 3, \dots\}. The 12th term is

f(12) = 51

. The value

63

occurs at the 15th term.

Check: Verify

f(15) = 4(15) + 3 = 60 + 3 = 63

✓. Verify

f(1) = 4(1) + 3 = 7

✓, matching the first term given.

Worked example

Writing a Geometric Sequence as a Discrete Function

A sequence begins:

5, 15, 45, 135, \dots

(a) Identify the type of sequence and find the common ratio. (b) Write the sequence as a discrete function

f(n)

. (c) Evaluate

f(6)

. (d) Is the value

1215

a term of this sequence? If yes, which term?

Find the common ratio
Divide each term by the one before it: $\frac{15}{5} = 3$ , $\frac{45}{15} = 3$ , $\frac{135}{45} = 3$ . The ratio is constant at $3$ , confirming this is a geometric sequence with first term $a = 5$ and common ratio $r = 3$ .
$r = 3, a = 5$
Write the general function rule
Use the geometric sequence formula $f(n) = a \cdot r^{n-1}$ . Substitute $a = 5$ and $r = 3$ . State the domain explicitly.
$f(n) = 5 · 3^{n-1}, n ∈ \{1, 2, 3, \dots\}$
Verify the first few terms
Check $f(1) = 5 \cdot 3^{0} = 5 \cdot 1 = 5$ ✓. Check $f(2) = 5 \cdot 3^{1} = 15$ ✓. Check $f(3) = 5 \cdot 3^{2} = 5 \cdot 9 = 45$ ✓. The formula matches the given terms.
$f(1) = 5, f(2) = 15, f(3) = 45$
Evaluate f(6)
Substitute $n = 6$ into the formula. Compute the exponent first: $3^{5} = 243$ . Then multiply by $5$ .
$f(6) = 5 · 3^{5} = 5 · 243 = 1215$
Determine whether 1215 is a term
From the calculation above, $f(6) = 1215$ . So $1215$ is indeed a term of the sequence — specifically, the 6th term. Because $n = 6$ is a positive integer, this is a valid answer within the sequence's domain.
$f(6) = 1215 \implies n = 6$

Answer: The sequence is geometric. The function is

f(n) = 5 \cdot 3^{n-1}

, n ∈ \{1, 2, 3, \dots\}. The 6th term is

f(6) = 1215

. Yes,

1215

is the 6th term of the sequence.

Check: List the terms: CAD 5, 15, 45, 135, 405, 1215. Counting to the 6th position gives

1215

✓.

Common mistakes and how to avoid them

Connecting the dots on a sequence graph with a solid line or curve.

Correction: Plot only isolated dots at integer values of

n

. A sequence is discrete, so non-integer inputs do not exist and must not be shown.

Using

n = 0

as the first term number.

Correction: In MCR3U, the domain of a sequence starts at

n = 1

. The first term is

t_1 = f(1)

, not

f(0)

. Using

n = 0

shifts every term number and produces the wrong formula.

Forgetting to check that a solved term number

n

is a positive integer.

Correction: When solving

f(n) = k

, the answer is only valid if

n

is a positive integer. A fractional or negative solution means

k

is not actually a term of the sequence.

Applying the arithmetic formula to a geometric sequence, or vice versa.

Correction: Always check the pattern first. Compute consecutive differences to test for arithmetic; compute consecutive ratios to test for geometric. Use the matching formula.

Writing

f(n) = a \cdot r^{n}

instead of

f(n) = a \cdot r^{n-1}

for a geometric sequence.

Correction: The exponent must be

n - 1

so that substituting

n = 1

gives

f(1) = a \cdot r^{0} = a

, which equals the first term. Using

r^n

would make

f(1) = ar

, which is the second term.

Lesson summary

A sequence is a discrete function because it assigns exactly one output (term value) to each input (term number), and the domain is restricted to positive integers.
The term number $n$ is the input and the term value $f(n)$ is the output; writing $f(n)$ instead of $t_n$ connects sequence notation directly to function notation.
An arithmetic sequence has a constant common difference $d$ and follows the linear rule $f(n) = a + (n-1)d$ .
A geometric sequence has a constant common ratio $r$ and follows the exponential rule $f(n) = a \cdot r^{n-1}$ .
Sequence graphs show only isolated dots at positive integer values of $n$ — never a connected line or curve.
Function notation makes it straightforward to find any specific term by evaluating $f(n)$ , or to find which term has a given value by solving $f(n) = k$ for a positive integer $n$ .

Check your understanding

Question 1

Which statement best explains why a sequence is called a discrete function?

Its outputs increase by the same amount each time.
Each term number (a positive integer) maps to exactly one term value.
Its graph is always a straight line through the origin.
It can only have a finite number of terms.

Show answer and explanation

Each term number (a positive integer) maps to exactly one term value.

A function requires each input to produce exactly one output. A sequence does this with positive integers as inputs, making it a discrete function. The other options describe specific types of sequences or incorrect ideas — sequences can be infinite and their graphs are not always straight lines.

Question 2

The function

f(n) = 6n - 2

with domain

\{1, 2, 3, \dots\}

represents a sequence. What is

f(5)

?

28
30
32
22

Show answer and explanation

28

Substitute

n = 5

:

f(5) = 6(5) - 2 = 30 - 2 = 28

. Option B forgets to subtract 2; option C adds 2 instead; option D uses

n = 4

instead of

n = 5

.

Question 3

A sequence is

4, 12, 36, 108, \dots

Which function rule describes it correctly?

$f(n) = 4 + 8(n-1)$
$f(n) = 4 \cdot 3^{n}$
$f(n) = 4 \cdot 3^{n-1}$
$f(n) = 3 \cdot 4^{n-1}$

Show answer and explanation

f(n) = 4 \cdot 3^{n-1}

Each term is multiplied by

3

, so the sequence is geometric with

a = 4

and

r = 3

. The correct formula is

f(n) = 4 \cdot 3^{n-1}

. Option A is arithmetic (constant difference), which does not fit. Option B uses

3^n

, giving

f(1) = 12

not

4

. Option D swaps

a

and

r

.

Question 4

For the sequence

f(n) = 5n + 2

, a student solves

f(n) = 19

and gets

n = 3.4

. What does this mean?

The 3rd and 4th terms both equal 19.
The value 19 is not a term of this sequence because 3.4 is not a positive integer.
The formula is wrong and needs to be recalculated.
The 4th term equals 19 because you round up to the nearest integer.

Show answer and explanation

The value 19 is not a term of this sequence because 3.4 is not a positive integer.

Solving

5n + 2 = 19

gives

5n = 17

, so

n = 3.4

. Since term numbers must be positive integers,

3.4

is not valid. The value

19

is simply not a term of this sequence. You never round

n

— only exact positive integers are valid term numbers.

Key terms

Sequence: An ordered list of numbers arranged according to a consistent rule or pattern. Each number in the list is called a term.
Term: An individual number in a sequence. The $n$ -th term is written $t_n$ or $f(n)$ .
Term number (index): The position of a term in a sequence, always a positive integer. It serves as the input to the sequence function.
Discrete function: A function whose domain consists of separated, individual values (here, positive integers) rather than all real numbers.
Arithmetic sequence: A sequence in which consecutive terms differ by a fixed constant called the common difference $d$ .
Geometric sequence: A sequence in which consecutive terms share a fixed constant ratio $r$ , found by dividing any term by the previous one.
Common difference ( $d$ ): The constant value added to each term of an arithmetic sequence to produce the next term.
Common ratio ( $r$ ): The constant value by which each term of a geometric sequence is multiplied to produce the next term.

Continue through MCR3U

View the complete MCR3U Ontario Grade 11 Mathematics curriculum and lessons

Tutor support for C1.1 · Connect sequences with discrete functions

Working on C1.1 · Connect sequences with discrete functions in MCR3U? Bring the question and your own attempt. In a live session, you can label the axes and connect each feature of the graph to the equation or situation. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث C1.1 · پیوند دنباله‌ها با تابع‌های گسسته در درس MCR3U، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید محورها را برچسب بزنید و هر ویژگی نمودار را به معادله یا موقعیت مسئله ربط دهید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation C1.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question