Arithmetic Progressions is one of the most important chapters in Class 10 Maths because it helps students understand patterns, sequences, formulas, and sum-based questions in a very structured way. This chapter is highly scoring in board exams because once the formulas and concepts are clear, most questions become direct and manageable.
Many students feel confused at the beginning because they see many formulas like common difference, aₙ, Sₙ, arithmetic mean, and term-based shortcuts all at once. But the chapter becomes much easier once students understand the basic idea behind an AP. The full chapter is built on one simple observation: the difference between consecutive terms remains constant.
At Deeksha Vedantu, we always encourage students to study Arithmetic Progressions in one sequence: first understand what an AP is, then learn common difference, then nth term, and finally move to sum of n terms and board-style questions. This makes the chapter much easier to revise and remember.
Why Arithmetic Progressions Is Important in Class 10
Arithmetic Progressions is an important algebra chapter and regularly contributes to board-exam marks.
Why Students Should Prepare This Chapter Well
- it is a regular board-exam chapter
- it includes direct formula-based questions
- it includes MCQs, short answers, and long answers
- it helps students improve sequence-based reasoning
- it includes both concept-based and application-based questions
Chapter Overview at a Glance
This quick table helps students revise the full chapter faster.
Quick Concept Table
| Topic | Key idea |
| Arithmetic Progression (AP) | A sequence in which the difference between consecutive terms remains constant |
| First term | The starting term of the AP |
| Common difference | The fixed difference between two consecutive terms |
| nth term | The term at the nth position |
| Sum of n terms | The total of the first n terms |
| Arithmetic mean | The middle term between two terms in AP |
What Is an Arithmetic Progression
An arithmetic progression, or AP, is a sequence of numbers in which the difference between any two consecutive terms is the same.
Simple Meaning
If the gap between one term and the next term remains constant throughout the sequence, then the sequence is called an arithmetic progression.
Examples of AP
| Sequence | Common difference |
| 2, 4, 6, 8, … | 2 |
| 100, 70, 40, 10, … | -30 |
| 5, 5, 5, 5, … | 0 |
Important Note
The common difference can be:
- positive
- negative
- zero
As long as it remains constant, the sequence is an AP.
Terms of an AP
In an arithmetic progression, each number is called a term.
Naming the Terms
| Position | Notation |
| First term | a₁ |
| Second term | a₂ |
| Third term | a₃ |
| Fourth term | a₄ |
| nth term | aₙ |
Common Difference
The constant difference between two consecutive terms of an AP is called the common difference.
Symbol of Common Difference
The common difference is denoted by d.
Formula for Common Difference
d = a₂ – a₁
It can also be written as:
- d = a₃ – a₂
- d = a₄ – a₃
Easy Rule
Always subtract the preceding term from the succeeding term.
Solved Example 1: Find the Common Difference
Find the common difference of:
10, 15, 20, 25, …
Given
Sequence:
10, 15, 20, 25, …
Solution
d = 15 – 10 = 5
Answer
The common difference is 5.
Solved Example 2: Negative Common Difference
Find the common difference of:
-3, -6, -9, -12, …
Given
Sequence:
-3, -6, -9, -12, …
Solution
d = -6 – (-3)
d = -6 + 3 = -3
Answer
The common difference is -3.
Arithmetic Mean
If three numbers a, b, and c are in AP, then the middle term is called the arithmetic mean of the first and third terms.
Important Relation
If a, b, c are in AP, then:
b – a = c – b
So:
2b = a + c
This is a very important result used in many board questions.
Arithmetic Mean Summary Table
| If three terms are in AP | Important result |
| a, b, c | 2b = a + c |
General Form of an AP
If a₁ is the first term and d is the common difference, then the AP can be written as:
a₁, a₁ + d, a₁ + 2d, a₁ + 3d, …
This is the general form of an arithmetic progression.
nth Term of an AP
The nth term of an AP is one of the most important formulas of the chapter.
Formula for nth Term
aₙ = a₁ + (n – 1)d
This formula helps students find any term directly without writing all the terms.
Why This Formula Matters
Using this formula, students can find:
- the 10th term
- the 20th term
- the nth term
- the position of a given term
all without expanding the full sequence.
Solved Example 3: Find the 10th Term
Find the 10th term of the AP:
3, 8, 13, 18, …
Given
- a₁ = 3
- d = 5
- n = 10
Step 1
Use the formula:
aₙ = a₁ + (n – 1)d
Step 2
Substitute the values:
a₁₀ = 3 + (10 – 1) × 5
Step 3
Simplify:
a₁₀ = 3 + 45 = 48
Answer
The 10th term is 48.
Solved Example 4: Find the Value of n
Which term of the AP 1, 5, 9, 13, … is 129?
Given
- a₁ = 1
- d = 4
- aₙ = 129
Step 1
Use the nth term formula:
aₙ = a₁ + (n – 1)d
Step 2
Substitute the values:
129 = 1 + (n – 1)4
Step 3
Simplify:
128 = 4(n – 1)
32 = n – 1
n = 33
Answer
129 is the 33rd term.
nth Term from the End
Sometimes students are asked to find the nth term from the last.
Formula
nth term from the end = l – (n – 1)d
Where:
- l = last term
- d = common difference
- n = position from the end
This formula is useful when the last term is known.
Important AP Shortcuts for 3, 4, and 5 Terms
These shortcuts are very useful in board questions.
Three Terms in AP
If three terms are in AP, they can be taken as:
a – d, a, a + d
Why This Is Useful
Their sum becomes:
(a – d) + a + (a + d) = 3a
So the d terms cancel.
Four Terms in AP
If four terms are in AP, they can be taken as:
a – 3d, a – d, a + d, a + 3d
This is a useful symmetric form in algebraic questions.
Five Terms in AP
If five terms are in AP, they can be taken as:
a – 2d, a – d, a, a + d, a + 2d
Why This Is Very Useful
Their sum becomes:
(a – 2d) + (a – d) + a + (a + d) + (a + 2d) = 5a
So the d terms cancel completely.
Shortcut Form Summary Table
| Number of terms | Standard AP form |
| 3 terms | a – d, a, a + d |
| 4 terms | a – 3d, a – d, a + d, a + 3d |
| 5 terms | a – 2d, a – d, a, a + d, a + 2d |
Solved Example 5: Three Terms in AP
If the sum of three numbers in AP is 30, find the middle term.
Given
Sum of three terms = 30
Step 1
Let the three terms be:
a – d, a, a + d
Step 2
Add them:
(a – d) + a + (a + d) = 30
Step 3
Simplify:
3a = 30
a = 10
Answer
The middle term is 10.
Solved Example 6: Five Terms in AP
The sum of five numbers in AP is 40. Find the middle term.
Given
Sum of five terms = 40
Step 1
Let the terms be:
a – 2d, a – d, a, a + d, a + 2d
Step 2
Add them:
5a = 40
Step 3
Simplify:
a = 8
Answer
The middle term is 8.
Sum of the First n Terms of an AP
This is the second major formula section of the chapter.
Formula Using First and Last Term
Sₙ = n/2 (a₁ + l)
Where:
- Sₙ = sum of first n terms
- a₁ = first term
- l = last term
Formula Using First Term and Common Difference
Sₙ = n/2 [2a₁ + (n – 1)d]
This is the most commonly used formula in Class 10.
When to Use Which Formula
| Situation | Formula to use |
| First term, common difference, and n are known | Sₙ = n/2 [2a₁ + (n – 1)d] |
| First term, last term, and n are known | Sₙ = n/2 (a₁ + l) |
Why the Sum Formula Is Important
It helps students find:
- the sum of first 10 terms
- the sum of first 20 terms
- the total saving, earning, or quantity after several steps
- application-based answers in word problems
Solved Example 7: Find the Sum of the First 20 Terms
Find the sum of the first 20 terms of the AP:
3, 7, 11, 15, …
Given
- a₁ = 3
- d = 4
- n = 20
Step 1
Use the formula:
Sₙ = n/2 [2a₁ + (n – 1)d]
Step 2
Substitute the values:
S₂₀ = 20/2 [2 × 3 + (20 – 1)4]
Step 3
Simplify:
S₂₀ = 10 [6 + 76]
S₂₀ = 10 × 82
S₂₀ = 820
Answer
The sum of the first 20 terms is 820.
Solved Example 8: Saving Money in AP
A student saves 32 rupees in the first month, 36 rupees in the second month, and 40 rupees in the third month. In how many months will the total saving become 2000 rupees?
Given
- a₁ = 32
- d = 4
- Sₙ = 2000
Step 1
Use the sum formula:
Sₙ = n/2 [2a₁ + (n – 1)d]
Step 2
Substitute the values:
2000 = n/2 [2 × 32 + (n – 1)4]
Step 3
Simplify:
2000 = n/2 [64 + 4n – 4]
2000 = n/2 (60 + 4n)
2000 = n(30 + 2n)
Step 4
Form the quadratic equation:
2n² + 30n – 2000 = 0
Divide by 2:
n² + 15n – 1000 = 0
Step 5
Factorise:
n² + 40n – 25n – 1000 = 0
n(n + 40) – 25(n + 40) = 0
(n – 25)(n + 40) = 0
Step 6
Take the valid value:
n = 25
Answer
The total saving becomes 2000 rupees in 25 months.
Important Question Types from Arithmetic Progressions
Board exams usually ask repeated question styles from this chapter.
Case 1: Find the Common Difference
These are direct and scoring.
Case 2: Check Whether a Sequence Is an AP
Students compare consecutive differences.
Case 3: Find the nth Term
These are very common.
Case 4: Find Which Term Has a Given Value
These are based on aₙ = a₁ + (n – 1)d.
Case 5: Find the Sum of First n Terms
These are based on Sₙ formulas.
Case 6: 3-Term, 4-Term, or 5-Term AP Questions
These use symmetric term shortcuts.
Case 7: Word Problems
These are based on savings, earnings, or pattern-based situations.
Case 8: Case-Based Questions
These combine formulas with interpretation.
Question Type Summary Table
| Case | Focus area |
| Case 1 | Common difference calculation |
| Case 2 | Constant difference check |
| Case 3 | Direct use of nth term formula |
| Case 4 | Position-finding using aₙ |
| Case 5 | Sum formula application |
| Case 6 | Shortcut forms for AP terms |
| Case 7 | Equation formation from situations |
| Case 8 | Interpretation and application |
Common Mistakes Students Make in Arithmetic Progressions
Common Mistakes Table
| Mistake | Correct idea |
| Subtracting in the wrong order | Always do succeeding term minus preceding term |
| Confusing aₙ and Sₙ | aₙ is a term, while Sₙ is the sum of terms |
| Using a₁ + nd instead of a₁ + (n – 1)d | The nth term formula is aₙ = a₁ + (n – 1)d |
| Forgetting to reject impossible values | Word problems may not allow negative answers |
| Mixing up last term and nth term | Read carefully whether the question asks for l, aₙ, or Sₙ |
Quick Revision Sheet for Arithmetic Progressions
This section is useful before board exams.
Quick Revision Table
| Topic | Formula or idea |
| Common difference | d = a₂ – a₁ |
| nth term | aₙ = a₁ + (n – 1)d |
| Sum of n terms | Sₙ = n/2 [2a₁ + (n – 1)d] |
| Sum using last term | Sₙ = n/2 (a₁ + l) |
| Three terms in AP | a – d, a, a + d |
| Five terms in AP | a – 2d, a – d, a, a + d, a + 2d |
| Arithmetic mean condition | 2b = a + c |
Best Study Strategy for Arithmetic Progressions
Arithmetic Progressions becomes much easier when revised in a fixed order.
Step-by-Step Revision Table
| Step | What to do |
| Step 1 | Understand AP through the idea of constant difference |
| Step 2 | Memorise the core formulas |
| Step 3 | Practise 3-term and 5-term shortcuts |
| Step 4 | Solve pattern-based word problems |
| Step 5 | Revise formula and concept together |
Practice Questions
This section helps students revise through standard board-style questions.
Important Practice Questions
- Find the common difference of the AP:
7, 11, 15, 19, …
- Find the 12th term of the AP:
5, 9, 13, 17, …
- Which term of the AP 3, 8, 13, 18, … is 98?
- The sum of three numbers in AP is 24. Find the middle term.
- The sum of five numbers in AP is 55. Find the middle term.
- Find the sum of the first 15 terms of the AP:
2, 5, 8, 11, …
- A student saves 50 rupees in the first month, 60 in the second, 70 in the third, and so on. In how many months will the total saving become 2750 rupees?
FAQs
Q1. What is an arithmetic progression in Class 10 Maths?
An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is always the same.
Q2. What is the common difference in an AP?
The common difference is the constant difference between consecutive terms and is written as d.
Q3. What is the formula for the nth term of an AP?
The formula is aₙ = a₁ + (n – 1)d.
Q4. What is the formula for the sum of n terms of an AP?
The formula is Sₙ = n/2 [2a₁ + (n – 1)d].
Q5. What are three terms in AP usually taken as?
They are usually taken as a – d, a, and a + d.
Q6. What are five terms in AP usually taken as?
They are usually taken as a – 2d, a – d, a, a + d, and a + 2d.
Q7. What is arithmetic mean in AP?
If a, b, c are in AP, then b is called the arithmetic mean of a and c.
Q8. How can I score well in Arithmetic Progressions?
You can score well by understanding the meaning of AP, memorising the formulas, practising 3-term and 5-term shortcuts, and solving sum-based questions regularly.
Conclusion
Arithmetic Progressions is one of the most direct and scoring chapters in Class 10 Maths because it is built on a simple idea: constant difference. Once students understand that one concept properly, the formulas for nth term and sum of n terms become much easier to use.
The best way to prepare this chapter is to revise the formulas with meaning, practise shortcut forms like 3-term and 5-term AP, and solve a variety of board-style questions. At Deeksha Vedantu, we always remind students that AP becomes easy when the pattern is understood clearly and the formulas are applied step by step.
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