Arithmetic Sequences

Learn to identify, analyze, and work with arithmetic sequences where terms increase or decrease by a constant difference.

Advanced25 minLesson
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Definition

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is always the same. This constant difference is called the common difference and is denoted by dd.
For example: 3,7,11,15,19,…3, 7, 11, 15, 19, \ldots
Here, each term is 4 more than the previous term, so d=4d = 4.

The Explicit Formula

To find any term in an arithmetic sequence, use the explicit formula:
an=a1+(n−1)⋅da_n = a_1 + (n - 1) \cdot d
Where:
  • ana_n = the nnth term (the term we want to find)
  • a1a_1 = the first term
  • nn = the position of the term
  • dd = the common difference

Finding the Common Difference

To find dd, subtract any term from the term that follows it:
d=an+1−and = a_{n+1} - a_n

Try it now

Which sequence is arithmetic?

Worked Examples

Find the common difference for the sequence: 8,5,2,−1,−4,…8, 5, 2, -1, -4, \ldots
1

Choose two consecutive terms

Let's use a1=8a_1 = 8 and a2=5a_2 = 5 → First two terms identified
2

Subtract the first from the second

d=a2−a1=5−8=−3d = a_2 - a_1 = 5 - 8 = -3 → d=−3d = -3
3

Verify with another pair

2−5=−32 - 5 = -3 and −1−2=−3-1 - 2 = -3 → Confirmed: d=−3d = -3

Common Mistakes

Using nn instead of (n−1)(n-1) in the formula

Why it's wrong: The formula an=a1+nda_n = a_1 + nd would give you the (n+1)(n+1)th term, not the nnth term. When n=1n=1, we should get a1a_1, which requires (n−1)=0(n-1) = 0.

Correct: Always use an=a1+(n−1)⋅da_n = a_1 + (n-1) \cdot d. Check: when n=1n=1, a1=a1+0⋅d=a1a_1 = a_1 + 0 \cdot d = a_1.

Subtracting in the wrong order when finding dd

Why it's wrong: Students sometimes calculate an−an+1a_n - a_{n+1} instead of an+1−ana_{n+1} - a_n, which gives the wrong sign for dd.

Correct: Always subtract the earlier term from the later term: d=an+1−and = a_{n+1} - a_n

Forgetting that dd can be negative

Why it's wrong: Arithmetic sequences can decrease! A sequence like 20,17,14,11,…20, 17, 14, 11, \ldots has d=−3d = -3.

Correct: Check if the sequence is increasing (d>0d > 0) or decreasing (d<0d < 0) before writing the formula.

Confusing the term number with the term value

Why it's wrong: In a5=23a_5 = 23, the subscript 5 is the position (5th term), and 23 is the value.

Correct: ana_n means the term at position nn. The subscript is always the position.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Arithmetic Sequences Explained: nth Term and Partial Sums

by Understand The Math

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Interactive Visual

Sequence Explorer

3,
7,
11,
15,
19,
23,
27,
31
, ...

Formula

a_n = 3 + 4(n - 1) = 4n - 1
Sum Formula: S_8 = 8/2 × (2×3 + 7×4) = 136
12345678
a_1

3

a_5

19

a_10

39

a_20

79

to

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Practice Problems

17 problems
Problem 1 of 17
Easy
Which sequence is arithmetic?

Why It Matters

Arithmetic sequences appear everywhere in daily life and are fundamental to understanding patterns:
  • Savings Plans: If you save 50 dollars each month, your total savings form an arithmetic sequence: 50, 100, 150, 200, ...
  • Seating Arrangements: Theater rows often increase by a fixed number of seats: 20, 24, 28, 32, ...
  • Depreciation: A car losing 2000 dollars in value each year follows an arithmetic pattern
  • Stair Construction: Each step rises by the same height, forming an arithmetic sequence of elevations
Understanding arithmetic sequences leads to the study of arithmetic series (sums) and prepares you for more complex patterns like geometric sequences.

Real World Applications

Savings and Financial Planning

Regular savings deposits create arithmetic sequences. Understanding these patterns helps with financial planning.

Example:

If you save 75 euros per month starting from 0, after nn months you have 75n75n euros. If you start with 200 euros, your balance forms the sequence 200,275,350,425,…200, 275, 350, 425, \ldots

1Try It Yourself

You open a savings account with 500 euros and deposit 125 euros every month.

How much will you have after 2 years (24 months)?

Step 1: Write the mathematical expression

Use an=a1+(n−1)⋅da_n = a_1 + (n-1) \cdot d with a1=500a_1 = 500 and d=125d = 125

Construction and Architecture

Builders use arithmetic sequences when planning staircases, seating, and structural patterns.

Example:

A staircase has a total rise of 3 meters over 15 steps. Each step rises by 315=0.2\frac{3}{15} = 0.2 meters. The heights form the sequence 0.2,0.4,0.6,…,3.00.2, 0.4, 0.6, \ldots, 3.0 meters.

2Try It Yourself

A pyramid of cans has 1 can on top, 3 in the second row, 5 in the third row, and so on.

How many cans are in the 10th row?

Step 1: Write the mathematical expression

This is an arithmetic sequence with a1=1a_1 = 1 and d=2d = 2

Sports and Fitness

Training programs often increase intensity following arithmetic patterns.

Example:

A runner starts with 2 km and adds 0.5 km each week. After nn weeks, they run 2+0.5(n−1)2 + 0.5(n-1) km. Week 10: 2+0.5×9=6.52 + 0.5 \times 9 = 6.5 km.

3Try It Yourself

A gym membership costs 30 euros to join plus 15 euros per month.

What is the total cost after 12 months?

Step 1: Write the mathematical expression

First month total is 45 euros, then add 15 each month

Key Takeaways

  • 1An arithmetic sequence has a constant difference dd between consecutive terms
  • 2The explicit formula is an=a1+(n−1)⋅da_n = a_1 + (n-1) \cdot d
  • 3Find dd by subtracting any term from the next: d=an+1−and = a_{n+1} - a_n
  • 4The common difference dd can be positive (increasing sequence) or negative (decreasing sequence)
  • 5To find a specific term, substitute the position nn into the explicit formula

Frequently Asked Questions

A sequence is a list of numbers following a pattern (e.g., 2,5,8,11,…2, 5, 8, 11, \ldots). A series is the sum of the terms in a sequence (e.g., 2+5+8+11+…2 + 5 + 8 + 11 + \ldots). We study arithmetic series after mastering sequences.
A sequence is a list of numbers following a pattern (e.g., 2,5,8,11,…2, 5, 8, 11, \ldots). A series is the sum of the terms in a sequence (e.g., 2+5+8+11+…2 + 5 + 8 + 11 + \ldots). We study arithmetic series after mastering sequences.
Yes! If d=0d = 0, every term is the same: 5,5,5,5,…5, 5, 5, 5, \ldots is technically an arithmetic sequence with d=0d = 0.
Calculate the differences between consecutive terms. If all differences are equal, it is arithmetic. For 3,7,11,153, 7, 11, 15: 7−3=47-3=4, 11−7=411-7=4, 15−11=415-11=4. All differences are 4, so it is arithmetic with d=4d=4.
Use the formula to set up equations. If a3=10a_3 = 10 and a7=22a_7 = 22, then a1+2d=10a_1 + 2d = 10 and a1+6d=22a_1 + 6d = 22. Subtract to get 4d=124d = 12, so d=3d = 3.

Glossary

Arithmetic sequence
A sequence where each term differs from the previous term by a constant amount called the common difference
Common difference
The constant value dd added to each term to get the next term in an arithmetic sequence
Explicit formula
A formula that directly calculates any term: an=a1+(n−1)⋅da_n = a_1 + (n-1) \cdot d
Term
Each number in a sequence; ana_n represents the term at position nn
First term
The starting value of the sequence, denoted a1a_1
nth term
The term at position nn in the sequence, denoted ana_n

Formula Card

Explicit Formula

an=a1+(n−1)⋅da_n = a_1 + (n-1) \cdot d

Find the $n$th term directly

Common Difference

d=an+1−and = a_{n+1} - a_n

Difference between consecutive terms

Alternative Form

an=am+(n−m)⋅da_n = a_m + (n-m) \cdot d

Find $a_n$ using any known term $a_m$

Today's challengeOne problem, three tries

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