Arithmetic Progression — CBSE Class 10 Formula Explained Clearly

The Formula aₙ = a + (n − 1)d Tracks Repeated Addition

Learn what each term represents before attempting sequence problems.

By Dhruv Chandravanshi | Class 10 Mathematics | CBSE

Have You Ever Noticed That Some Patterns Just Keep Adding the Same Number — Over and Over?

In this article, you will learn:

• What an arithmetic progression actually is — understood as a pattern before becoming a formula
• What a, d, n, and aₙ represent — not as symbols alone but as real values inside a sequence
• How the nth term formula works and why it is structured the way it is
• What CBSE Class 10 expects in AP questions and the exact mistakes that usually cost marks

By the end, you will understand that the AP formula is not something to memorise blindly.

It is a formula you could derive yourself once you recognise the pattern it describes.

A Pattern Hidden in Rows of Chairs

Arjun’s school in Patna organises a prize distribution ceremony every year.

The chairs are arranged in rows.

• Row 1 → 5 chairs
• Row 2 → 8 chairs
• Row 3 → 11 chairs
• Row 4 → 14 chairs

Each row contains three more chairs than the previous row.

The person arranging the chairs does not recount every row individually.

He simply adds 3 chairs each time.

Arjun sits in the 5th row.

He wonders how many chairs are in his row.

His teacher looks once at the arrangement and immediately says:

17 chairs.

No counting.

Just recognising the pattern.

Understanding the Pattern

Look carefully at the sequence:

5, 8, 11, 14, 17 …

Each term increases by 3.

Now rewrite the sequence differently:

• Row 1 → 5
• Row 2 → 5 + 1 × 3
• Row 3 → 5 + 2 × 3
• Row 4 → 5 + 3 × 3
• Row 5 → 5 + 4 × 3

Notice something important.

The multiplier of 3 is always one less than the row number.

For the 5th row:

5 + (5 − 1) × 3

= 5 + 12

= 17

Now try the 100th row:

5 + (100 − 1) × 3

= 5 + 297

= 302

The same structure works for every row.

This predictable pattern is called an Arithmetic Progression.

What Is an Arithmetic Progression?

An Arithmetic Progression (AP) is a sequence of numbers where the difference between consecutive terms remains constant.

That fixed difference is called the common difference.

Example:

5, 8, 11, 14, 17 …

Differences:

• 8 − 5 = 3
• 11 − 8 = 3
• 14 − 11 = 3

Since the difference remains the same, the sequence forms an AP.

Important Symbols in Arithmetic Progression

Five symbols appear repeatedly in AP problems.

a

The first term of the sequence.

Example: 5

d

The common difference.

Example: 3

n

The position number of the term.

Example:

• 5th term
• 20th term
• 100th term

aₙ

The value of the nth term.

Sₙ

The sum of the first n terms.

The Nth Term Formula

The formula for the nth term of an AP is:

aₙ = a + (n − 1)d

This formula calculates any term directly without listing the entire sequence.

Using the chair example:

• a = 5
• d = 3
• n = 5

a₅ = 5 + (5 − 1) × 3

= 5 + 12

= 17

Why the Formula Uses (n − 1)

The formula contains (n − 1) for a reason.

The first term already exists.

To reach the second term, we add d once.

To reach the third term, we add d twice.

So by the nth term, d has been added exactly (n − 1) times.

That is why:

aₙ = a + (n − 1)d

How to Identify an AP

Step 1 — Check the Differences

Example:

3, 7, 11, 15, 19 …

Differences:

• 7 − 3 = 4
• 11 − 7 = 4
• 15 − 11 = 4

The difference is constant.

So the sequence is an AP.

Step 2 — Identify a and d

Example:

5, 8, 11, 14 …

• a = 5
• d = 3

Step 3 — Apply the Formula

aₙ = a + (n − 1)d

Substitute the values and solve.

Sum of an Arithmetic Progression

Sometimes we need the sum of several terms instead of one specific term.

The formula is:

Sₙ = n/2 × [2a + (n − 1)d]

Another useful form:

Sₙ = n/2 × (a + l)

where l represents the last term.

This form is useful when the first and last terms are already known.

Worked Examples

Example 1 — Find a Specific Term

AP:

7, 11, 15, 19 …

Find the 20th term.

• a = 7
• d = 4
• n = 20

a₂₀ = 7 + (20 − 1) × 4

= 7 + 76

= 83

Example 2 — Find the Position of a Term

AP:

5, 8, 11, 14 …

Which term equals 50?

• a = 5
• d = 3

50 = 5 + (n − 1) × 3

45 = 3(n − 1)

n − 1 = 15

n = 16

So 50 is the 16th term.

Example 3 — Find the Sum of Terms

AP:

2, 5, 8, 11 …

Find the sum of the first 15 terms.

• a = 2
• d = 3
• n = 15

S₁₅ = 15/2 × [2×2 + 14×3]

= 15/2 × (4 + 42)

= 15/2 × 46

= 345

Visualising Arithmetic Progressions

Arithmetic progressions appear whenever a quantity changes by the same amount repeatedly.

Real-Life Examples of AP

Monthly Savings

A student saves:

• January → ₹200
• February → ₹250
• March → ₹300

Each month increases by ₹50.

This forms an AP.

Ladder Rungs

Suppose ladder rungs shorten by 2 cm each step:

45 cm, 43 cm, 41 cm …

This is also an AP with a negative common difference.

Seating Arrangements

Theatre rows and stadium seating often increase by a fixed number of seats.

Architects frequently use AP calculations while planning such layouts.

Common Mistakes Students Make

Mistake 1 — Confusing d with the Second Term

Example:

5, 8, 11, 14

• Second term = 8
• Common difference = 8 − 5 = 3

Using 8 instead of 3 produces incorrect answers.

Mistake 2 — Mixing the Term Formula with the Sum Formula

aₙ finds a single term.

Sₙ finds the sum of multiple terms.

Using the wrong formula leads to mistakes.

Mistake 3 — Thinking AP Must Always Increase

An AP can also decrease.

Example:

20, 17, 14, 11 …

Here:

d = −3

The sequence is still an AP.

Important AP Formulas for CBSE

aₙ = a + (n − 1)d

Sₙ = n/2 × [2a + (n − 1)d]

Sₙ = n/2 × (a + l)

Types of Questions CBSE Often Asks

• Find the nth term of an AP
• Find the position of a term
• Find the sum of the first n terms
• Word problems involving seating, savings, and distances
• Determine whether a sequence forms an AP

Core Idea in Simple Words

An arithmetic progression is a sequence where the same number is added repeatedly.

The first term is a.

The fixed addition is d.

To reach the nth term, d is added exactly (n − 1) times.

That produces the formula:

aₙ = a + (n − 1)d

Practice Questions

1. The first term of an AP is 3 and the common difference is 4. Write the first five terms and find the 25th term.
2. Which term of the AP 7, 13, 19, 25… equals 205?
3. Find the sum of the first 20 terms of the AP 4, 9, 14, 19…
4. The sum of three consecutive terms in an AP is 27 and their product is 504. Find the terms.
5. The 7th term of an AP is 32 and the 13th term is 62. Find the first term and common difference.

Frequently Asked Questions

What is the difference between a sequence and a series?

A sequence lists numbers in order.

A series represents the sum of those numbers.

Can the common difference be negative?

Yes. A decreasing sequence can still form an AP.

Can the common difference be a fraction?

Yes.

Example:

1, 1.5, 2, 2.5 …

Here:

d = 0.5

What happens if d = 0?

All terms remain equal:

a, a, a, a …

Why does the sum formula contain n/2?

Because pairs of terms from opposite ends add to the same value, creating n/2 equal pairs.

Related Topics

• Quadratic Equations
• Real Numbers and Number Patterns
• Statistics and Mean

This article follows the CBSE Class 10 Mathematics syllabus (Chapter 5 — Arithmetic Progressions) and aligns with NCERT mathematics concepts. All formulas and examples reflect standard CBSE examination patterns.

About the Author

Dhruv Chandravanshi writes about physical and logical systems — from motion to matrices — examining the structures that govern how numbers and matter behave.