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 .
For example:
Here, each term is 4 more than the previous term, so .

The Explicit Formula

To find any term in an arithmetic sequence, use the explicit formula:
Where:
  • = the th term (the term we want to find)
  • = the first term
  • = the position of the term
  • = the common difference

Finding the Common Difference

To find , subtract any term from the term that follows it:

Try it now

Which sequence is arithmetic?

Worked Examples

Find the common difference for the sequence:

1

Choose two consecutive terms

Let's use and → First two terms identified

2

Subtract the first from the second

→

3

Verify with another pair

and → Confirmed:

Common Mistakes

Using instead of in the formula

Why it's wrong: The formula would give you the th term, not the th term. When , we should get , which requires .

Correct: Always use . Check: when , .

Subtracting in the wrong order when finding

Why it's wrong: Students sometimes calculate instead of , which gives the wrong sign for .

Correct: Always subtract the earlier term from the later term:

Forgetting that can be negative

Why it's wrong: Arithmetic sequences can decrease! A sequence like has .

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

Confusing the term number with the term value

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

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

Interactive Visual

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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 months you have euros. If you start with 200 euros, your balance forms the sequence

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 with and

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 meters. The heights form the sequence 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 and

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 weeks, they run km. Week 10: 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 between consecutive terms
  • 2The explicit formula is
  • 3Find by subtracting any term from the next:
  • 4The common difference can be positive (increasing sequence) or negative (decreasing sequence)
  • 5To find a specific term, substitute the position into the explicit formula

Frequently Asked Questions

A sequence is a list of numbers following a pattern (e.g., ). A series is the sum of the terms in a sequence (e.g., ). We study arithmetic series after mastering sequences.
A sequence is a list of numbers following a pattern (e.g., ). A series is the sum of the terms in a sequence (e.g., ). We study arithmetic series after mastering sequences.
Yes! If , every term is the same: is technically an arithmetic sequence with .
Calculate the differences between consecutive terms. If all differences are equal, it is arithmetic. For : , , . All differences are 4, so it is arithmetic with .
Use the formula to set up equations. If and , then and . Subtract to get , so .

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 added to each term to get the next term in an arithmetic sequence
Explicit formula
A formula that directly calculates any term:
Term
Each number in a sequence; represents the term at position
First term
The starting value of the sequence, denoted
nth term
The term at position in the sequence, denoted

Formula Card

Explicit Formula

Find the $n$th term directly

Common Difference

Difference between consecutive terms

Alternative Form

Find $a_n$ using any known term $a_m$

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