Develop and use general-term formulas for arithmetic and geometric sequences

Ontario Grade 11 Mathematics

Discrete Functions

Building and applying the $t_n$ formula for both types of sequences

Sequences appear whenever quantities change by a predictable rule — weekly savings, the bouncing height of a ball, or the number of bacteria doubling every hour. Being able to jump directly to any term, without listing every term before it, is a powerful skill. This lesson builds the two most important general-term formulas in MCR3U: one for arithmetic sequences (add the same amount each time) and one for geometric sequences (multiply by the same amount each time). You will see where each formula comes from, practise using it, and learn to spot the errors that most students make.

What you will learn

Identify whether a sequence is arithmetic, geometric, or neither by examining its pattern.
Develop the general-term formula for an arithmetic sequence and explain why it works.
Develop the general-term formula for a geometric sequence and explain why it works.
Use each formula to find any term or to find an unknown value such as a missing first term or common ratio.
Solve problems that require choosing the correct formula and interpreting the result in context.

Prerequisite Bridge: What Is a Sequence?

A sequence is an ordered list of numbers. Each number in the list is called a term. The first term is written

t_1

, the second term

t_2

, and so on. The term in position

n

is written

t_n

. Think of

n

as the term number — it is always a positive integer.

You already know from Grade 10 that linear relationships have a constant rate of change, and that exponential relationships involve repeated multiplication. Those two ideas are exactly what drive arithmetic and geometric sequences respectively. Recognising which pattern you are dealing with is always your first job.

A term is one number in the sequence; its position is its term number $n$ .
$t_n$ means the term at position $n$ .
Arithmetic sequences are linked to linear growth; geometric sequences are linked to exponential growth.

Arithmetic Sequences and the General-Term Formula

An arithmetic sequence is formed by adding the same fixed number to each term to get the next one. That fixed number is called the common difference and is written

d

. For example,

3, 7, 11, 15, \ldots

has

t_1 = 3

and

d = 4

because each term is 4 more than the one before it. You can always verify

d

by subtracting any term from the term that follows it:

d = t_2 - t_1 = t_3 - t_2

, and so on.

To find any term without listing every previous term, notice the pattern:

t_1 = t_1

,

t_2 = t_1 + d

,

t_3 = t_1 + 2d

,

t_4 = t_1 + 3d

. The number of times

d

is added is always one less than the term number. So the general-term formula for an arithmetic sequence is:

This formula works because you start at

t_1

and add

d

exactly

(n-1)

times to reach term number

n

. Every arithmetic general-term formula is a linear expression in

n

, which connects directly to the linear functions you studied in Grade 10.

You can use the formula in several ways: find a specific term given

t_1

and

d

; find

t_1

if you know a term and

d

; find

d

if you know two terms; or determine which term number has a given value.

t_n = t_1 + (n-1)d

Common difference: $d = t_{n} - t_{n-1}$ (subtract any term from the next one).
The general-term formula is $t_n = t_1 + (n-1)d$ .
$(n-1)$ counts how many times $d$ has been added since the first term.
The formula gives a linear expression in $n$ , matching linear growth.
You can solve for $t_1$ , $d$ , or $n$ by substituting known values and rearranging.

Geometric Sequences and the General-Term Formula

A geometric sequence is formed by multiplying each term by the same fixed number to get the next one. That fixed number is called the common ratio and is written

r

. For example,

5, 10, 20, 40, \ldots

has

t_1 = 5

and

r = 2

because each term is twice the previous one. You can find

r

by dividing any term by the term before it:

r = t_2 \div t_1 = t_3 \div t_2

, and so on.

The same pattern-building approach works here:

t_1 = t_1

,

t_2 = t_1 \cdot r

,

t_3 = t_1 \cdot r^2

,

t_4 = t_1 \cdot r^3

. The exponent on

r

is always one less than the term number. So the general-term formula for a geometric sequence is:

This formula works because you start at

t_1

and multiply by

r

exactly

(n-1)

times. Every geometric general-term formula involves an exponential expression in

n

, matching the exponential functions you met in Grade 10.

The common ratio

r

can be a fraction (terms shrink), a negative number (terms alternate in sign), or a number greater than 1 (terms grow). As long as

r \neq 0

and

r \neq 1

, the sequence is genuinely geometric.

t_n = t_1 · r^{n-1}

Common ratio: $r = t_{n} \div t_{n-1}$ (divide any term by the previous one).
The general-term formula is $t_n = t_1 \cdot r^{n-1}$ .
The exponent $(n-1)$ counts how many multiplications have occurred since the first term.
The formula gives an exponential expression in $n$ , matching exponential growth or decay.
$r$ can be positive, negative, or a fraction, leading to growing, alternating, or shrinking sequences.

Connecting the Two Formulas: Similarities and Differences

Both formulas share the same structure: start with the first term, then apply the rule

(n-1)

times. For arithmetic sequences the rule is repeated addition of

d

, giving

t_1 + (n-1)d

. For geometric sequences the rule is repeated multiplication by

r

, giving

t_1 \cdot r^{n-1}

. Keeping this parallel in mind prevents mixing up the two formulas.

A quick way to decide which formula applies: calculate the differences between consecutive terms. If the differences are equal, the sequence is arithmetic. If the differences are not equal, calculate the ratios of consecutive terms instead. If those ratios are equal, the sequence is geometric. If neither differences nor ratios are constant, neither formula applies and you need a different approach.

One important special case: if

r = 1

every term equals

t_1

, so the sequence is constant (and could also be seen as arithmetic with

d = 0

). These edge cases rarely appear on assessments, but recognising them shows strong understanding.

Both formulas use $(n-1)$ because the first term requires zero applications of the rule.
Check constant differences for arithmetic; check constant ratios for geometric.
Arithmetic $\rightarrow$ linear expression in $n$ ; Geometric $\rightarrow$ exponential expression in $n$ .

Applying the Formulas: Strategy and Context

When a problem gives you the sequence directly, identify

t_1

and either

d

or

r

first, write the formula, then substitute. When a problem gives you two terms but not the first term, set up two equations using the general-term formula and solve. For arithmetic sequences this usually means subtracting one equation from the other. For geometric sequences it usually means dividing one equation by the other to isolate

r

.

Context problems often ask for the term number

n

rather than the term value. After substituting and simplifying, you will need to solve a linear equation (arithmetic) to find

n

. Always check that

n

is a positive whole number — if it is not, the value you were asked about is not actually in the sequence.

A useful habit: after finding your answer, substitute it back into the original formula to confirm the result. This self-check catches arithmetic slips before they cost marks.

Always identify the sequence type before selecting a formula.
For two unknown quantities, write two equations and solve the system.
The answer for $n$ must be a positive integer; otherwise the value is not a term of the sequence.
Substitute your answer back into the formula to verify.

Arithmetic vs. Geometric Sequences at a Glance

Feature	Arithmetic Sequence	Geometric Sequence
Rule between terms	Add a fixed number $d$	Multiply by a fixed number $r$
How to find the rule	$d = t_n - t_{n-1}$	$r = t_n \div t_{n-1}$
General-term formula	$t_n = t_1 + (n-1)d$	$t_n = t_1 \cdot r^{n-1}$
Type of expression in $n$	Linear	Exponential
Example sequence	$2, 5, 8, 11, \ldots$ ( $d = 3$ )	$2, 6, 18, 54, \ldots$ ( $r = 3$ )

Worked example

Arithmetic Sequence: Finding a Term and a Term Number

An arithmetic sequence begins

-5, 1, 7, 13, \ldots

(a) Write the general-term formula. (b) Find

t_{20}

. (c) Determine which term of the sequence equals

103

.

Identify the first term and common difference
Read $t_1$ directly from the sequence. Then find $d$ by subtracting the first term from the second: $d = 1 - (-5) = 6$ . Verify with the next pair: $7 - 1 = 6$ . The difference is constant, confirming this is arithmetic.
$t_1 = -5, d = 6$
Write the general-term formula
Substitute $t_1 = -5$ and $d = 6$ into the arithmetic formula $t_n = t_1 + (n-1)d$ , then simplify the expression so it is in slope-intercept form.
$t_n = -5 + (n-1)(6) = 6n - 11$
Find the 20th term
Substitute $n = 20$ into the simplified formula $t_n = 6n - 11$ .
$t_{20} = 6(20) - 11 = 120 - 11 = 109$
Find which term equals 103
Set $t_n = 103$ and solve for $n$ using the simplified formula $6n - 11 = 103$ . Add $11$ to both sides, then divide by $6$ .
$6n = 114 \implies n = 19$
Verify both answers
Check $t_{20}$ : substituting $n = 20$ gives $6(20) - 11 = 109$ . Check $n = 19$ : $6(19) - 11 = 114 - 11 = 103$ . Both match, so the answers are confirmed.
$t_{19} = 6(19) - 11 = 103 \checkmark$

Answer: The general-term formula is

t_n = 6n - 11

. The 20th term is

t_{20} = 109

. The term that equals

103

is the 19th term.

Check: Using

t_n = 6n - 11

:

t_{19} = 6(19) - 11 = 103

✓ and

t_{20} = 6(20) - 11 = 109

✓.

Worked example

Geometric Sequence: Finding the Formula from Two Non-Consecutive Terms

In a geometric sequence, the 2nd term is

12

and the 5th term is

324

. (a) Find the common ratio

r

and the first term

t_1

. (b) Write the general-term formula. (c) Find

t_7

.

Write two equations using the geometric formula
The general-term formula is $t_n = t_1 \cdot r^{n-1}$ . Write one equation for $n = 2$ and one for $n = 5$ , using the given term values. t_1 r = 12 and t_1 r^4 = 324
Divide the equations to eliminate $t_1$
Dividing the equation for $t_5$ by the equation for $t_2$ cancels $t_1$ and leaves a single equation in $r$ only. This works because both sides of the second equation are exactly $r^3$ times the corresponding sides of the first.
$\frac{t_1 r^4}{t_1 r} = \frac{324}{12} \implies r^3 = 27$
Solve for $r$
Take the cube root of both sides. Since $27 = 3^3$ , the cube root is exactly $3$ .
$r = \sqrt[3]{27} = 3$
Solve for $t_1$
Substitute $r = 3$ back into the equation $t_1 r = 12$ and divide both sides by $3$ .
$t_1 = \frac{12}{3} = 4$
Write the general-term formula
Substitute $t_1 = 4$ and $r = 3$ into $t_n = t_1 \cdot r^{n-1}$ .
$t_n = 4 · 3^{n-1}$
Find the 7th term
Substitute $n = 7$ into the formula. Calculate $3^6 = 729$ first, then multiply by $4$ .
$t_7 = 4 · 3^{6} = 4 · 729 = 2916$
Verify $t_2$ and $t_5$ using the formula
Check that the formula reproduces the given information. $t_2 = 4 \cdot 3^1 = 12$ ✓ and $t_5 = 4 \cdot 3^4 = 4 \cdot 81 = 324$ ✓.
$t_2 = 4 · 3^1 = 12 \checkmark, t_5 = 4 · 3^4 = 324 \checkmark$

Answer: The common ratio is

r = 3

, the first term is

t_1 = 4

, the general-term formula is

t_n = 4 \cdot 3^{n-1}

, and the 7th term is

t_7 = 2916

.

Check: Listing terms: CAD 4, 12, 36, 108, 324, 972, 2916. Term 2 is

12

✓, term 5 is

324

✓, term 7 is

2916

✓.

Common mistakes and how to avoid them

Writing

t_n = t_1 + nd

instead of

t_n = t_1 + (n-1)d

, adding

d

one too many times.

Correction: The first term already exists before any addition. You only add

d

a total of

(n-1)

times to reach term

n

. Check: when

n=1

, the formula must return

t_1

, so

(n-1)

must equal zero at

n=1

.

Writing

t_n = t_1 \cdot r^n

instead of

t_n = t_1 \cdot r^{n-1}

, multiplying by

r

one extra time.

Correction: The same reasoning applies: the first term requires zero multiplications. Substitute

n=1

to check — the formula must give

t_1 \cdot r^0 = t_1

, not

t_1 \cdot r

.

Using the arithmetic formula for a geometric sequence (or vice versa) without first checking the type of sequence.

Correction: Always test for constant differences first. If differences are not constant, test for a constant ratio. Choose the correct formula only after confirming the sequence type.

Accepting a non-integer or negative answer for the term number

n

without questioning it.

Correction: The term number

n

must be a positive whole number. If your equation gives

n = 7.5

or

n = -3

, the target value is not a term in that sequence.

Dividing the smaller-term equation by the larger-term equation and getting a fractional

r^k

with the wrong exponent, leading to an incorrect value of

r

.

Correction: Divide the equation with the higher power of

r

by the equation with the lower power. Check the exponent on

r

in the result: it should equal the difference in the two term numbers (e.g., terms 5 and 2 differ by 3, so

r^3

appears).

Lesson summary

An arithmetic sequence has a constant difference $d$ between consecutive terms; its general-term formula is $t_n = t_1 + (n-1)d$ , a linear expression in $n$ .
A geometric sequence has a constant ratio $r$ between consecutive terms; its general-term formula is $t_n = t_1 \cdot r^{n-1}$ , an exponential expression in $n$ .
In both formulas the factor $(n-1)$ appears because the first term requires zero applications of the rule.
When two non-consecutive terms are given, write two equations and divide (geometric) or subtract (arithmetic) to find the unknown values.
The term number $n$ must always be a positive integer; verify every answer by substituting back into the formula.
Checking whether differences or ratios are constant is the reliable way to identify the sequence type before choosing a formula.

Check your understanding

Question 1

Which formula correctly gives the general term of the arithmetic sequence

10, 6, 2, -2, \ldots

?

$t_n = 10 + 4n$
$t_n = 10 - 4(n-1)$
$t_n = 10 \cdot (-4)^{n-1}$
$t_n = -4n + 10$

Show answer and explanation

t_n = 10 - 4(n-1)

The first term is

t_1 = 10

and the common difference is

d = 6 - 10 = -4

. Substituting into

t_n = t_1 + (n-1)d

gives

t_n = 10 + (n-1)(-4) = 10 - 4(n-1)

. Option A adds

4n

instead of subtracting and misses the

(n-1)

structure. Option C wrongly uses the geometric formula with a negative base. Option D simplifies to

-4n + 10

, which gives

t_1 = -4(1)+10 = 6 \neq 10

, so it is incorrect.

Question 2

A geometric sequence has

t_1 = 2

and

r = -3

. What is

t_4

?

$-54$
$54$
$-162$
$18$

Show answer and explanation

-54

Using

t_n = t_1 \cdot r^{n-1}

:

t_4 = 2 \cdot (-3)^3 = 2 \cdot (-27) = -54

. The exponent is

4-1=3

, not

4

. Option B forgets the negative sign. Option C uses the exponent

4

instead of

3

. Option D corresponds to

t_3 = 2 \cdot (-3)^2 = 18

, which is the wrong term.

Question 3

In an arithmetic sequence,

t_3 = 20

and

t_7 = 44

. What is the common difference

d

?

$4$
$6$
$8$
$12$

Show answer and explanation

6

The two equations are

t_1 + 2d = 20

and

t_1 + 6d = 44

. Subtracting the first from the second gives

4d = 24

, so

d = 6

. Option A gives

4d = 16

, meaning

t_7 - t_3 = 16

, not

24

. Options C and D also fail the subtraction check.

Question 4

A geometric sequence begins

3, 6, 12, 24, \ldots

Which term of this sequence equals

384

?

Term 6
Term 7
Term 8
Term 9

Show answer and explanation

Term 8

Here

t_1 = 3

and

r = 2

, so

t_n = 3 \cdot 2^{n-1}

. Setting

3 \cdot 2^{n-1} = 384

gives

2^{n-1} = 128 = 2^7

, so

n - 1 = 7

and

n = 8

. Checking:

t_8 = 3 \cdot 2^7 = 3 \cdot 128 = 384

✓.

Key terms

Sequence: An ordered list of numbers that follow a rule. Each number in the list is called a term.
Term ( $t_n$ ): One number in a sequence. The subscript $n$ gives its position; for example, $t_3$ is the third term.
Arithmetic sequence: A sequence in which each term is found by adding the same fixed number (the common difference) to the previous term.
Common difference ( $d$ ): The fixed number added to each term of an arithmetic sequence to produce the next term. Found by computing $t_n - t_{n-1}$ .
Geometric sequence: A sequence in which each term is found by multiplying the previous term by the same fixed number (the common ratio).
Common ratio ( $r$ ): The fixed number by which each term of a geometric sequence is multiplied to produce the next term. Found by computing $t_n \div t_{n-1}$ .
General-term formula: A formula that gives the value of any term $t_n$ directly from its position number $n$ , without needing to list all previous terms.
Term number ( $n$ ): The position of a term in a sequence. It must always be a positive whole number (positive integer).

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation C2.2. 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.

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