C2.1 Classify arithmetic, geometric, and other sequences

Ontario Grade 11 Mathematics

Discrete Functions

MCR3U · Expectation C2.1

Have you ever noticed that a gym membership charges an extra CAD 15 every month, or that a bacterial colony doubles every hour? Both situations produce lists of numbers that follow a rule. In mathematics, an ordered list of numbers that follows a rule is called a sequence. Before you can work with formulas or sums for sequences, you need to be able to look at a list and decide what kind of pattern it follows. That classification step is exactly what this lesson is about. You will learn to sort sequences into three categories — arithmetic, geometric, or neither — using two simple tests that only require subtraction and division.

What you will learn

Identify whether a sequence is arithmetic, geometric, or neither by testing for a common difference or common ratio.
Calculate the common difference of an arithmetic sequence and the common ratio of a geometric sequence.
Recognize sequences that are neither arithmetic nor geometric and describe their patterns.
Apply classification skills to sequences given in list form, in a table, or in a real-world context.

Prerequisite Bridge: What You Already Know

In Grade 10 you worked with linear relations (constant rate of change) and exponential growth (repeated multiplication). Those ideas connect directly to sequences. A sequence that grows by adding the same amount each time is the sequence version of a linear relation. A sequence that grows by multiplying by the same number each time is the sequence version of exponential growth. Keeping those connections in mind will make classification feel natural rather than like brand-new material.

You also know how to divide fractions and simplify ratios. You will use those skills when you check whether consecutive terms share a common ratio. Make sure you are comfortable writing a ratio such as

\frac{18}{6} = 3

before moving on.

Linear relations → arithmetic sequences (constant addition).
Exponential relations → geometric sequences (constant multiplication).
You only need subtraction and division to classify most sequences.

Arithmetic Sequences: The Addition Pattern

An arithmetic sequence is a sequence in which the same number is added (or subtracted) from one term to get the next. That fixed amount is called the common difference, written

d

. To find

d

, subtract any term from the term that follows it.

Consider the sequence

5, 12, 19, 26, 33, \ldots

Subtract each term from the one after it:

12 - 5 = 7

,

19 - 12 = 7

,

26 - 19 = 7

,

33 - 26 = 7

. Every difference equals

7

, so

d = 7

and the sequence is arithmetic.

The common difference can be negative. The sequence

20, 15, 10, 5, 0, \ldots

has

d = -5

because you subtract

5

each time. A negative common difference simply means the sequence is decreasing. The classification test is the same: check whether all consecutive differences are equal.

Why does this test work? If every gap between neighbours is the same, the sequence is growing (or shrinking) at a perfectly steady rate — exactly like a linear relation. The moment even one gap is different from the others, the sequence cannot be arithmetic.

d = t_{n+1} - t_n

Arithmetic sequences have a constant common difference $d$ .
Find $d$ by computing (next term) $-$ (current term) for every consecutive pair.
If all differences are equal, the sequence is arithmetic.
A negative $d$ means the sequence decreases; the classification rule is unchanged.

Geometric Sequences: The Multiplication Pattern

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by the same fixed number. That fixed multiplier is called the common ratio, written

r

. To find

r

, divide any term by the term that comes before it.

Consider the sequence

3, 6, 12, 24, 48, \ldots

Divide each term by the one before it:

\frac{6}{3} = 2

,

\frac{12}{6} = 2

,

\frac{24}{12} = 2

,

\frac{48}{24} = 2

. Every ratio equals

2

, so

r = 2

and the sequence is geometric.

The common ratio can be a fraction. The sequence

81, 27, 9, 3, 1, \ldots

has

r = \frac{1}{3}

because each term is one-third of the previous one. When

0 < r < 1

, the terms get smaller but the sequence is still geometric.

It is important to apply the ratio test to every consecutive pair, not just the first two. One matching ratio is not enough evidence — you must confirm the ratio is the same throughout the sequence. If any consecutive ratio differs from the others, the sequence is not geometric.

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

Geometric sequences have a constant common ratio $r$ .
Find $r$ by computing $\frac{t_{n+1}}{t_n}$ for every consecutive pair.
If all ratios are equal, the sequence is geometric.
A fractional $r$ (where $0 < r < 1$ ) gives a decreasing geometric sequence.

Sequences That Are Neither Arithmetic nor Geometric

Not every sequence fits neatly into one of the two categories above. A sequence is classified as neither arithmetic nor geometric when the consecutive differences are not all equal AND the consecutive ratios are not all equal.

Consider the sequence

1, 1, 2, 3, 5, 8, 13, \ldots

The differences are CAD 0, 1, 1, 2, 3, 5 — not constant. The ratios are

1, 2, 1.5, 1.\overline{6}, 1.6

— not constant. This is the famous Fibonacci sequence, where each new term is the sum of the two terms before it. It is a valid mathematical sequence with a clear rule, but that rule is neither repeated addition nor repeated multiplication.

Another example of a 'neither' sequence is

1, 4, 9, 16, 25, \ldots

(the perfect squares). Differences: CAD 3, 5, 7, 9 — not constant. Ratios:

4, 2.25, 1.\overline{7}, 1.5625

— not constant. The rule here is that the

n

-th term equals

n^2

, which is a different kind of pattern entirely.

When you meet a sequence that is neither arithmetic nor geometric, your job is to describe whatever pattern you do observe — such as 'each term is the square of its position' or 'each term is the sum of the two before it.' You do not need to force it into a category that does not fit.

If differences are not all equal AND ratios are not all equal, the sequence is neither arithmetic nor geometric.
Always test both conditions before labelling a sequence 'neither.'
Describe the actual rule you notice for sequences that do not fit either category.
Examples: perfect squares, Fibonacci-type sequences, alternating-sign patterns.

A Classification Strategy You Can Always Use

Here is a reliable step-by-step approach you can apply to any sequence. First, list the consecutive differences. If they are all equal, stop — the sequence is arithmetic, and

d

is that common difference. If the differences are not all equal, list the consecutive ratios instead. If all ratios are equal, the sequence is geometric, and

r

is that common ratio. If neither test passes, the sequence is neither arithmetic nor geometric, and you should describe whatever pattern you can identify.

One important caution: always use at least three consecutive terms (giving you at least two differences or two ratios) before deciding. A single pair of terms always produces exactly one difference and one ratio, so one pair alone can never confirm a pattern.

Keeping a short table of your difference or ratio calculations beside the sequence is a great habit. It makes your reasoning visible and helps you catch arithmetic errors before they lead to a wrong classification.

Step 1: Compute all consecutive differences. Equal? → Arithmetic.
Step 2: Compute all consecutive ratios. Equal? → Geometric.
Step 3: Neither test passes? → Neither; describe the observed pattern.
Always use at least three terms (two gaps) before classifying.

Quick Classification Reference

Sequence Type	Test to Apply	What Must Be True	Label and Value to State
Arithmetic	Compute $t_{n+1} - t_n$ for all consecutive pairs	All differences are equal	State $d$ = common difference
Geometric	Compute $t_{n+1} \div t_n$ for all consecutive pairs	All ratios are equal	State $r$ = common ratio
Neither	Both tests above fail	Differences not constant AND ratios not constant	Describe the observed pattern in words

Worked example

Classifying Three Sequences Given as Lists

Classify each sequence as arithmetic, geometric, or neither. For arithmetic sequences state

d

; for geometric sequences state

r

.
(a)

100, 85, 70, 55, 40, \ldots

(b)

2, 6, 18, 54, 162, \ldots

(c)

0, 1, 3, 6, 10, 15, \ldots

Test sequence (a): compute consecutive differences
Subtract each term from the one after it: $85 - 100 = -15$ , $70 - 85 = -15$ , $55 - 70 = -15$ , $40 - 55 = -15$ . Every difference equals $-15$ , which is constant, so the sequence is arithmetic.
$d = -15$
Test sequence (b): compute consecutive differences first
Subtract each term from the one after it: $6 - 2 = 4$ , $18 - 6 = 12$ , $54 - 18 = 36$ . The differences CAD 4, 12, 36 are not equal, so the sequence is not arithmetic. Now test the ratios.
Test sequence (b): compute consecutive ratios
Divide each term by the one before it: $\frac{6}{2} = 3$ , $\frac{18}{6} = 3$ , $\frac{54}{18} = 3$ , $\frac{162}{54} = 3$ . Every ratio equals $3$ , which is constant, so the sequence is geometric.
$r = 3$
Test sequence (c): compute consecutive differences
Subtract each term from the one after it: $1 - 0 = 1$ , $3 - 1 = 2$ , $6 - 3 = 3$ , $10 - 6 = 4$ , $15 - 10 = 5$ . The differences CAD 1, 2, 3, 4, 5 are not equal, so the sequence is not arithmetic.
Test sequence (c): compute consecutive ratios
Divide each term by the one before it. The second term divided by the first is $\frac{1}{0}$ , which is undefined — so the ratio test cannot even be completed. The sequence is not geometric. It is neither. Notice the pattern: the differences increase by $1$ each time, meaning these are the triangular numbers.

Answer: (a) Arithmetic,

d = -15

. (b) Geometric,

r = 3

. (c) Neither (triangular numbers; differences increase by 1 each step).

Check: For (a): starting at

100

and repeatedly adding

-15

gives CAD 100, 85, 70, 55, 40

✓. For (b): starting at

2 and repeatedly multiplying by

3

gives CAD 2, 6, 18, 54, 162 ✓. For (c): confirming the differences CAD 1, 2, 3, 4, 5 are strictly increasing, not constant, and no common ratio exists ✓.

Worked example

Identifying a Sequence Type from a Real-World Table

A savings account is tracked over five months. The balances recorded are shown in the table below (in CAD): Month 1: CAD 500, Month 2: CAD 550, Month 3: CAD 605, Month 4: CAD 665.50, Month 5: CAD 732.05. A second account simply receives a CAD 60 top-up each month, starting at CAD 500: Month 1: CAD 500, Month 2: CAD 560, Month 3: CAD 620, Month 4: CAD 680, Month 5: CAD 740. Classify each account's balance sequence and state

d

or

r

where applicable.

Account 2 — test consecutive differences
Subtract each balance from the next: $560 - 500 = 60$ , $620 - 560 = 60$ , $680 - 620 = 60$ , $740 - 680 = 60$ . All differences equal $60$ . The sequence is arithmetic.
$d = 60$
Account 1 — test consecutive differences
Subtract each balance from the next: $550 - 500 = 50$ , $605 - 550 = 55$ , $665.50 - 605 = 60.50$ , $732.05 - 665.50 = 66.55$ . The differences CAD 50, 55, 60.50, 66.55 are not equal, so the sequence is not arithmetic. Move to the ratio test.
Account 1 — test consecutive ratios
Divide each balance by the previous one: $\frac{550}{500} = 1.1$ , $\frac{605}{550} = 1.1$ , $\frac{665.50}{605} = 1.1$ , $\frac{732.05}{665.50} = 1.1$ . Every ratio equals $1.1$ , which is constant. The sequence is geometric.
$r = 1.1$
Interpret the common ratio
A common ratio of $r = 1.1$ means the balance grows by 10% each month. This matches compound interest behaviour — the interest earned each month is itself a percentage of a growing balance, so the absolute increase gets larger every month even though the percentage stays the same.

Answer: Account 2 is arithmetic with

d = 60

(CAD 60 added each month). Account 1 is geometric with

r = 1.1

(balance grows by 10% each month).

Check: Account 2:

500 + 4(60) = 500 + 240 = 740

✓ matches Month 5. Account 1:

500 \times (1.1)^4 = 500 \times 1.4641 = 732.05

✓ matches Month 5.

Common mistakes and how to avoid them

Checking only the first two terms and assuming the whole sequence fits a pattern.

Correction: Always check every consecutive pair. One matching difference or ratio is not enough — you need all gaps (or all ratios) to be equal before classifying.

Subtracting in the wrong order, for example computing

t_n - t_{n+1}

instead of

t_{n+1} - t_n

.

Correction: Always subtract the earlier term from the later term. Doing it backwards gives the opposite sign for

d

and leads to a wrong answer.

Forgetting to test for geometric when the difference test fails, and labelling the sequence 'neither' too quickly.

Correction: Run both tests every time. Only after both tests fail can you correctly label a sequence as neither arithmetic nor geometric.

Concluding a sequence is geometric when the ratio test cannot be completed (for example, because a term is zero).

Correction: Division by zero is undefined. If any term is zero, the ratio

t_{n+1} \div t_n

cannot be computed for that pair, so the sequence cannot be geometric.

Confusing

d

and

r

: stating a ratio when the sequence is arithmetic, or a difference when it is geometric.

Correction:

d

is always found by subtraction and belongs to arithmetic sequences.

r

is always found by division and belongs to geometric sequences. Label each one clearly.

Lesson summary

A sequence is an ordered list of numbers that follows a rule; classifying it is the first step toward working with its formula or sum.
An arithmetic sequence has a constant common difference $d$ , found by subtracting any term from the term that follows it.
A geometric sequence has a constant common ratio $r$ , found by dividing any term by the term that precedes it.
Always test all consecutive pairs — not just the first two — before reaching a classification decision.
If both tests fail (differences not constant and ratios not constant), the sequence is neither arithmetic nor geometric; describe the pattern you observe.
Real-world sequences such as regular deposits (arithmetic) and compound-interest growth (geometric) can be classified using the same two tests.

Check your understanding

Question 1

What is the common difference of the arithmetic sequence

-3, 2, 7, 12, 17, \ldots

?

$d = -5$
$d = 3$
$d = 5$
$d = 7$

Show answer and explanation

d = 5

Subtract any term from the next:

2 - (-3) = 5

,

7 - 2 = 5

,

12 - 7 = 5

. All differences equal

5

, so

d = 5

.

Question 2

A sequence begins

4, 12, 36, 108, \ldots

What type of sequence is it, and what is the value of

r

?

Arithmetic with $d = 8$
Geometric with $r = 4$
Geometric with $r = 3$
Neither arithmetic nor geometric

Show answer and explanation

Geometric with

r = 3

Check differences:

12 - 4 = 8

,

36 - 12 = 24

— not equal, so not arithmetic. Check ratios:

\frac{12}{4} = 3

,

\frac{36}{12} = 3

,

\frac{108}{36} = 3

— all equal, so the sequence is geometric with

r = 3

.

Question 3

A student checks only the first two terms of a sequence, finds a difference of

6

, and immediately calls it arithmetic. What error did the student make?

Used division instead of subtraction to find the difference.
Did not convert the terms to decimals first.
Only checked one pair of terms instead of all consecutive pairs.
Should have checked the ratio first, not the difference.

Show answer and explanation

Only checked one pair of terms instead of all consecutive pairs.

You must verify that every consecutive pair of terms has the same difference. A single matching gap does not confirm the pattern holds throughout the sequence.

Question 4

Which sequence is NEITHER arithmetic NOR geometric?

$10, 20, 30, 40, 50, \ldots$
$3, 9, 27, 81, 243, \ldots$
$1, 2, 4, 7, 11, 16, \ldots$
$50, 25, 12.5, 6.25, \ldots$

Show answer and explanation

1, 2, 4, 7, 11, 16, \ldots

For CAD 1, 2, 4, 7, 11, 16: differences are CAD 1, 2, 3, 4, 5 — not constant. Ratios are

2, 2, 1.75, 1.571\ldots

— not constant. Both tests fail, so it is neither. The other three sequences are arithmetic (option A,

d=10

), geometric (option B,

r=3

), and geometric (option D,

r=0.5

).

Key terms

Sequence: An ordered list of numbers arranged according to a rule. Each number in the list is called a term.
Term: One individual number in a sequence. The first term is written $t_1$ , the second $t_2$ , and so on.
Arithmetic sequence: A sequence in which the same number is added to each term to produce the next term.
Common difference ( $d$ ): The fixed amount added from one term to the next in an arithmetic sequence. Found by computing $t_{n+1} - t_n$ .
Geometric sequence: A sequence in which each term is multiplied by the same fixed number to produce the next term.
Common ratio ( $r$ ): The fixed multiplier applied from one term to the next in a geometric sequence. Found by computing $t_{n+1} \div t_n$ .
Consecutive terms: Two terms that appear directly next to each other in a sequence, such as $t_3$ and $t_4$ .
Neither arithmetic nor geometric: A label for a sequence in which the consecutive differences are not all equal and the consecutive ratios are not all equal.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation C2.1. It is a study resource, not an official curriculum publication.

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C2.1 · Classify arithmetic, geometric, and other sequences

Ontario Grade 11 Mathematics

MCR3U · Expectation C2.1

What you will learn

Prerequisite Bridge: What You Already Know

Arithmetic Sequences: The Addition Pattern

Geometric Sequences: The Multiplication Pattern

Sequences That Are Neither Arithmetic nor Geometric

A Classification Strategy You Can Always Use

Quick Classification Reference

Classifying Three Sequences Given as Lists

Identifying a Sequence Type from a Real-World Table

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