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Arithmetic Progressions · Lesson 1 of 4

Arithmetic Progressions

“It is the only place in life where adding the exact same problem to your day every single step of the way is considered actual progress.”

Learning Objectives

• Recognize patterns in lists of numbers. • Understand what an Arithmetic Progression (AP) is. • Identify the first term and common difference of an AP. • Check whether a given sequence forms an AP. • Distinguish between finite and infinite Arithmetic Progressions.

Introduction

Look around you and you will notice patterns everywhere. Stairs rise by almost the same height, salaries may increase by a fixed amount every year, and someone saving the same extra amount each month creates a predictable pattern. Mathematics gives us a way to describe such patterns clearly.

Patterns Help Us Predict

A pattern becomes especially interesting when numbers change according to a fixed rule. If we know the rule, we can often predict numbers that have not even been written yet.

A Simple Example

Suppose a person earns ₹20,000 per month in the first year of a job, and the monthly salary increases by ₹2,000 every year.

Salary PatternLaTeX
Every new salary is obtained by adding ₹2,000 to the previous salary.
YearMonthly SalaryChange
1st year₹20,000—
2nd year₹22,000+ ₹2,000
3rd year₹24,000+ ₹2,000
4th year₹26,000+ ₹2,000
5th year₹28,000+ ₹2,000
What Should You Notice?

The important thing is not that these numbers represent salaries. The important thing is that the same number, 2000, is added every time. This repeated fixed change is the main idea behind an Arithmetic Progression.

The fixed change does not have to increase the numbers. Consider the lengths 45 cm, 43 cm, 41 cm, 39 cm, ... . Each new length is 2 cm smaller than the previous one.

A Decreasing PatternLaTeX
Subtracting 2 is the same as adding −2.

Not Every Pattern Is the Same

Many sequences follow patterns, but they do not all follow the same type of rule. Some repeatedly add a number, some multiply by a number, while others may follow completely different rules.

SequencePatternArithmetic Progression?
2, 5, 8, 11, ...Add 3 each timeYes
20, 15, 10, 5, ...Subtract 5 each timeYes
4, 4, 4, 4, ...Add 0 each timeYes
1, 2, 4, 8, ...Multiply by 2 each timeNo
1, 1, 2, 3, 5, ...Different changes occurNo
The Pattern We Will Study

In this chapter, we are interested in one special kind of number pattern: a pattern in which the same number is added to every term to obtain the next term.

Arithmetic Progression

Consider the sequence 5, 8, 11, 14, 17, ... . Starting from 5, we repeatedly add 3. Because the change between every pair of consecutive terms is the same, this sequence is an Arithmetic Progression.

Definition
Arithmetic Progression (AP)

An Arithmetic Progression, or AP, is a list of numbers in which each term after the first is obtained by adding the same fixed number to the preceding term.

Definition
Term

Each number appearing in an Arithmetic Progression is called a term of the AP.

Terms of an APLaTeX
Here 7 is the first term, 11 is the second term, 15 is the third term and so on.
Notation for Terms

We usually write the first term as a₁, the second term as a₂, the third term as a₃, and in general the nth term as aₙ.

Term NotationLaTeX

The First Term

Definition
First Term

The first number in an Arithmetic Progression is called its first term. It is usually represented by the letter a.

Worked Example: Finding the First Term

Problem
Find the first term of the AP 12, 17, 22, 27, ... .

  1. 1.The first number in the AP is 12.
  2. 2.Therefore, the first term a = 12.

The Common Difference

What makes an Arithmetic Progression special is that the difference between consecutive terms always remains the same.

Definition
Common Difference

The fixed number added to each term to obtain the next term is called the common difference of the AP. It is represented by d.

Finding the Common DifferenceLaTeX
Always subtract the earlier term from the term immediately after it.
Worked Example: Finding a Positive Common Difference

Problem
Find the first term and common difference of 6, 10, 14, 18, ... .

  1. 1.The first term is a = 6.
  2. 2.Subtract the first term from the second term.
  3. 3.d = 10 − 6 = 4.
  4. 4.Check the next pair: 14 − 10 = 4.
  5. 5.The difference remains the same.
  6. 6.Therefore, a = 6 and d = 4.
Checking the DifferenceLaTeX
Every consecutive difference is 4.

The Common Difference Can Be Positive, Negative or Zero

Positive Common Difference

If d is positive, the terms increase as we move through the AP. For example, 3, 8, 13, 18, ... has d = 5.

Negative Common Difference

If d is negative, the terms decrease as we move through the AP. For example, 20, 16, 12, 8, ... has d = −4.

Zero Common Difference

If d = 0, every term remains the same. For example, 7, 7, 7, 7, ... is also an Arithmetic Progression.

APdType of dBehaviour
3, 7, 11, 15, ...4PositiveIncreasing
18, 13, 8, 3, ...−5NegativeDecreasing
6, 6, 6, 6, ...0ZeroConstant
Common Mistake: Subtracting in the Wrong Order

When finding d, always calculate later term − earlier term. For the AP 10, 7, 4, 1, ... write d = 7 − 10 = −3, not 10 − 7 = 3.

Worked Example: A Decreasing AP

Problem
Find a and d for 15, 11, 7, 3, ... .

  1. 1.The first term is a = 15.
  2. 2.d = 11 − 15.
  3. 3.Therefore, d = −4.
  4. 4.Check: 7 − 11 = −4 and 3 − 7 = −4.
  5. 5.Hence, a = 15 and d = −4.

General Form of an Arithmetic Progression

Once we know the first term a and the common difference d, we can generate the entire Arithmetic Progression.

General Form of an APLaTeX
Each new term contains one more d than the term before it.
PositionTerm
1st terma
2nd terma + d
3rd terma + 2d
4th terma + 3d
Look Ahead

Notice an important pattern: to reach the second term, d is added once. To reach the third term, d is added twice. To reach the fourth term, d is added three times. This idea will become very important when we learn how to find the nth term in the next lesson.

Worked Example: Writing an AP from a and d

Problem
Write the first five terms of an AP whose first term is 4 and common difference is 3.

  1. 1.Here, a = 4 and d = 3.
  2. 2.First term = 4.
  3. 3.Second term = 4 + 3 = 7.
  4. 4.Third term = 7 + 3 = 10.
  5. 5.Fourth term = 10 + 3 = 13.
  6. 6.Fifth term = 13 + 3 = 16.
  7. 7.Therefore, the AP is 4, 7, 10, 13, 16, ... .
Worked Example: Writing an AP with Negative d

Problem
Write the first five terms of an AP when a = 10 and d = −3.

  1. 1.Start with the first term 10.
  2. 2.Add −3 to obtain each new term.
  3. 3.Second term = 10 + (−3) = 7.
  4. 4.Third term = 7 + (−3) = 4.
  5. 5.Fourth term = 4 + (−3) = 1.
  6. 6.Fifth term = 1 + (−3) = −2.
  7. 7.Therefore, the AP is 10, 7, 4, 1, −2, ... .

How Do We Check Whether a Sequence Is an AP?

A sequence is an Arithmetic Progression only if the difference between every pair of consecutive terms is the same. So the easiest test is to calculate consecutive differences and compare them.

AP TestLaTeX
If all consecutive differences are equal, the sequence is an AP.
Worked Example: Checking Whether a Sequence Is an AP

Problem
Check whether 4, 10, 16, 22, ... forms an Arithmetic Progression.

  1. 1.Find the difference between consecutive terms.
  2. 2.10 − 4 = 6.
  3. 3.16 − 10 = 6.
  4. 4.22 − 16 = 6.
  5. 5.All the consecutive differences are equal.
  6. 6.Therefore, the sequence is an AP with common difference d = 6.
VerificationLaTeX
The same difference appears every time.
Worked Example: A Sequence That Is Not an AP

Problem
Check whether 2, 4, 8, 16, ... is an Arithmetic Progression.

  1. 1.Calculate consecutive differences.
  2. 2.4 − 2 = 2.
  3. 3.8 − 4 = 4.
  4. 4.16 − 8 = 8.
  5. 5.The differences 2, 4 and 8 are not equal.
  6. 6.Therefore, this sequence is not an Arithmetic Progression.
Pattern Does Not Always Mean AP

A sequence may have an obvious pattern and still not be an Arithmetic Progression. For example, 2, 4, 8, 16, ... follows a pattern, but the differences are not equal. Therefore, it is not an AP.

Finding the Next Terms

Once we know that a sequence is an AP and know its common difference, finding the next terms is simple: keep adding d.

Worked Example: Finding the Next Terms

Problem
Find the next three terms of 5, 9, 13, 17, ... .

  1. 1.First find the common difference.
  2. 2.d = 9 − 5 = 4.
  3. 3.Add 4 to the last given term: 17 + 4 = 21.
  4. 4.Add 4 again: 21 + 4 = 25.
  5. 5.Add 4 again: 25 + 4 = 29.
  6. 6.Therefore, the next three terms are 21, 25 and 29.

Finite and Infinite Arithmetic Progressions

Definition
Finite AP

An Arithmetic Progression that contains a fixed number of terms and therefore has a last term is called a finite AP.

Example of a Finite APLaTeX
This AP stops at 25, so it contains only five terms.
Definition
Infinite AP

An Arithmetic Progression that continues without ending is called an infinite AP. It has no last term.

Example of an Infinite APLaTeX
The dots show that the pattern continues indefinitely.
SequenceTypeHas a Last Term?
2, 5, 8, 11, 14Finite APYes
10, 20, 30, 40, ...Infinite APNo
25, 20, 15, 10, 5, 0Finite APYes
7, 7, 7, 7, ...Infinite APNo

Arithmetic Progressions in Everyday Life

Arithmetic Progressions are useful whenever a quantity increases or decreases by the same amount repeatedly. Recognising this fixed change allows us to describe the situation mathematically.

SituationWhat Happens?Common Difference
A salary increases by ₹1,000 every yearFixed increase+1000
₹200 more is saved every monthFixed increase+200
Rows contain 2 fewer plants than the previous rowFixed decrease−2
A ladder rung becomes 3 cm shorter at every levelFixed decrease−3
The same amount is recorded every dayNo change0
A Useful Clue in Word Problems

Whenever you see phrases such as 'increases by the same amount', 'decreases uniformly', '₹50 more each time' or '3 less than the previous one', think about Arithmetic Progressions.

Common Mistakes

• Thinking that every number pattern is an AP. • Finding d by subtracting in the wrong order. • Forgetting that d may be negative. • Forgetting that d = 0 is allowed. • Confusing the first term a with the common difference d. • Assuming that an AP must always increase. • Checking only one pair of differences when the sequence has not already been confirmed to be an AP.

Quiz

Quick check

Which of the following is an Arithmetic Progression?

Quick check

What is the common difference of 10, 7, 4, 1, ... ?

Quick check

What does d represent in an Arithmetic Progression?

Quick check

Which statement about the common difference is correct?

Quick check

What is the next term of the AP 3, 6, 9, 12, ... ?

Quick check

Which is the general form of an Arithmetic Progression?

Quick check

Is 1, 2, 4, 8, 16, ... an Arithmetic Progression?

Practice Problems

Practice Questions
  1. Find the first term and common difference of 3, 7, 11, 15, ... .
  2. Find the first term and common difference of 20, 16, 12, 8, ... .
  3. Find the first term and common difference of 5, 5, 5, 5, ... .
  4. Write the first five terms of an AP with a = 2 and d = 4.
  5. Write the first five terms of an AP with a = 15 and d = −3.
  6. Check whether 7, 12, 17, 22, ... is an AP.
  7. Check whether 1, 4, 9, 16, ... is an AP.
  8. Find the next three terms of 11, 8, 5, 2, ... .
  9. State whether 4, 8, 12, 16, 20 is a finite or infinite AP.
  10. A student saves ₹100 in the first week and increases the amount saved by ₹25 every week. Write the amounts saved during the first five weeks.

Key Takeaways

Key Takeaways

• An Arithmetic Progression is a sequence in which the difference between consecutive terms remains constant. • Each number in an AP is called a term. • The first term is represented by a. • The common difference is represented by d. • The common difference can be positive, negative or zero. • We find the common difference using later term − earlier term. • The general form of an AP is a, a + d, a + 2d, a + 3d, ... . • A sequence is an AP only when its consecutive differences are equal. • A finite AP has a last term, while an infinite AP continues indefinitely.

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nth Term of an AP