Arithmetic Sequence Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Arithmetic Sequence.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A sequence where each term is obtained from the previous by adding a fixed constant called the common difference $d$ .

Add the same number each time — 2, 5, 8, 11,... (add 3 each step). This is constant-rate growth.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An arithmetic sequence increases by a fixed common difference $d$ every term, giving constant linear growth.

Common stuck point: The procedure for arithmetic sequence is the easy part; the trap is using $a_1+nd$ instead of $a_1+(n-1)d$ . Asking "Do I get the same number every time I subtract a term from the one after it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I get the same number every time I subtract a term from the one after it?

Worked Examples

Example 1

easy

An arithmetic sequence has

a_1 = 7

and

d = -3

. Find

a_{20}

and

S_{20}

Answer

a_{20} = -50

;

S_{20} = -430

First step

Use the arithmetic sequence formula to find the 20th term:

a_n = a_1 + (n-1)d

, where

a_1 = 7

d = -3

n = 20

Full solution

2
Calculate: $a_{20} = 7 + (20-1)(-3) = 7 - 57 = -50$
3
Apply the partial sum formula: $S_{20} = \frac{20}{2}(a_1 + a_{20}) = 10(7 + (-50)) = 10(-43) = -430$

With a negative common difference the sequence decreases. The sum formula averages the first and last terms and multiplies by the count.

Example 2

medium

In an arithmetic sequence

a_5 = 18

and

a_{12} = 46

. Find

a_1

and

d

Example 3

medium

Find the sum of the first 20 terms of the arithmetic sequence: 5, 8, 11, 14, ...

Example 4

easy

Write the general term

a_n

for

6, 10, 14, 18, \ldots

Example 5

hard

An auditorium has

30

rows. The first row has

18

seats and each row has

2

more seats than the previous. How many total seats?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the 15th term of

4, 9, 14, 19, \ldots

Example 2

medium

Find the sum of all integers from 1 to 100.

Example 3

easy

Find the common difference of

4, 9, 14, 19, \ldots

Example 4

easy

Find the 6th term of an arithmetic sequence with

a_1=3

and

d=4

Example 5

easy

The first term is 10 and

d=-3

. Find the 4th term.

Example 6

easy

2, 5, 8, 11

arithmetic or geometric?

Example 7

easy

Find the sum of the first 5 terms of

2, 4, 6, 8, 10

Example 8

easy

How many terms are there from

a_3

a_{10}

inclusive?

Example 9

easy

Write the general term for

7, 11, 15, 19, \ldots

Example 10

easy

The 1st term is 5 and the 2nd is 8. Find the 10th term.

Example 11

medium

The 3rd term of an arithmetic sequence is 14 and the 7th term is 30. Find

a_1

and

d

Example 12

medium

Find the sum of the first 20 positive even numbers

2+4+\cdots+40

Example 13

medium

How many terms of

5, 8, 11, \ldots

are needed to reach 50?

Example 14

medium

Find the sum

S=3+7+11+\cdots+99

Example 15

medium

In an arithmetic sequence,

a_5=20

and the common difference is 3. Find the sum of the first 5 terms.

Example 16

medium

The sum of the first

n

terms of

4, 7, 10, \ldots

is 175. Find

n

Example 17

challenge

Three numbers form an arithmetic sequence with sum 15 and product 80. Find them.

Example 18

challenge

Show that the sum of the first

n

odd numbers

1+3+5+\cdots+(2n-1)

equals

n^2

Example 19

challenge

The 4th term of an arithmetic sequence equals 3 times the 1st term, and the 6th term is 13. Find

a_1

and

d

Example 20

medium

The 2nd term is 11 and the 5th term is 23. Find the 8th term.

Example 21

medium

Find the sum

5+9+13+\cdots+45

Example 22

medium

An arithmetic sequence has

a_1=2

and

a_{10}=29

. Find

d

and the sum of the first 10 terms.

Example 23

easy

Find the common difference of

3, 7, 11, 15, \ldots

Example 24

easy

Find the 8th term of an arithmetic sequence with

a_1 = 2

and

d = 5

Example 25

easy

5, 10, 20, 40, \ldots

arithmetic or geometric?

Example 26

easy

Find the sum of

1 + 2 + 3 + \ldots + 50

Example 27

medium

Find the 25th term of

11, 15, 19, 23, \ldots

Example 28

medium

The 4th term of an arithmetic sequence is

17

and the 10th term is

41

. Find

a_1

and

d

Example 29

medium

Find the sum of the first

30

terms of

6, 10, 14, \ldots

Example 30

medium

How many terms of

7, 12, 17, \ldots

are needed to reach

82

Example 31

medium

Find the sum

S = 2 + 5 + 8 + \ldots + 59

Example 32

medium

Find the sum of the first

50

positive odd numbers.

Example 33

medium

The sum of the first

n

terms of

3, 7, 11, \ldots

190

. Find

n

Example 34

medium

Find the 12th term of an arithmetic sequence whose first three terms are

-2, 1, 4

Example 35

medium

An arithmetic sequence has

a_3 = 11

and

a_8 = 31

. Find

a_{15}

Example 36

medium

Find the sum

20 + 17 + 14 + \ldots + (-7)

Example 37

hard

Find the sum of all multiples of

3

between

1

and

200

Example 38

hard

The sum of the first

n

terms of an arithmetic sequence is

S_n = 2n^2 + 3n

. Find

a_1

and

d

Example 39

hard

a_1 + a_2 + a_3 = 24

and the three terms form an arithmetic sequence with common difference

5

, find them.

Example 40

hard

Find the number of terms in the sequence

-5, -2, 1, \ldots, 49

Example 41

challenge

Three numbers form an arithmetic sequence whose sum is

21

and whose product is

231

. Find them.

Example 42

challenge

The sum of the first

10

terms of an arithmetic sequence is

145

, and the sum of the next

10

terms is

445

. Find

a_1

and

d

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Related Concepts

Sequence Geometric Sequence

Background Knowledge

These ideas may be useful before you work through the harder examples.

sequence