Arithmetic Progression (AP) – nth Term & Sum of n Terms | Complete Guide for Class 10
Arithmetic Progression (AP) is one of the most scoring and conceptually simple chapters in Class 10 CBSE Mathematics. If you clearly understand the formula for the nth term and the sum of n terms, you can solve most exam questions confidently.
In this detailed guide, we will cover definitions, formulas, derivations, solved problems, word problems, and exam strategies related to Arithmetic Progression.
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What is an Arithmetic Progression (AP)?
An Arithmetic Progression is a sequence of numbers in which the difference between consecutive terms remains constant. This constant difference is called the Common Difference (d).
Examples:
- 2, 4, 6, 8, 10… (d = 2)
- 5, 9, 13, 17… (d = 4)
- 10, 7, 4, 1… (d = -3)
Important Terminology
- a = First term
- d = Common difference
- n = Number of terms
- aₙ = nth term
- Sₙ = Sum of first n terms
Formula for nth Term of AP
Watch: ARITHMETIC PROGRESSION NTH TERM BASIC PROBLEMS Explained (Step-by-Step)
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aₙ = a + (n − 1)d
This formula helps us find any term of the sequence without writing all previous terms.
Example 1: Find the 20th term of the AP: 3, 7, 11, 15…
a = 3, d = 4
a₂₀ = 3 + (20 − 1) × 4
= 3 + 19 × 4
= 3 + 76 = 79
20th term = 79
Formula for Sum of First n Terms
Sₙ = n/2 [2a + (n − 1)d]
Alternative formula:
Sₙ = n/2 (a + l) (where l is last term)
Example 2: Find the sum of first 10 terms of AP: 2, 5, 8…
a = 2, d = 3, n = 10
S₁₀ = 10/2 [2(2) + (10 − 1)3]
= 5 [4 + 27]
= 5 × 31 = 155
Sum = 155
Finding Common Difference
Common difference is calculated as:
d = a₂ − a₁
Example 3: Find d in AP: 7, 12, 17, 22…
d = 12 − 7 = 5
Word Problems on AP
Problem 4: Find the 15th term of AP: 4, 9, 14…
a = 4, d = 5
a₁₅ = 4 + (14 × 5)
= 4 + 70 = 74
Problem 5: How many terms of AP 3, 7, 11… are needed to get sum 406?
a = 3, d = 4
Sₙ = n/2 [2(3) + (n − 1)4] = 406
n/2 [6 + 4n − 4] = 406
n/2 [4n + 2] = 406
n(4n + 2) = 812
4n² + 2n − 812 = 0
Solve quadratic → n = 14
Problem 6: Find sum of first 50 natural numbers.
AP: 1, 2, 3…
a = 1, d = 1
S₅₀ = 50/2 [2 + 49]
= 25 × 51 = 1275
Application-Based Problems
Problem 7: A theatre has 20 rows. First row has 25 seats, each next row has 2 more seats. Find total seats.
a = 25, d = 2, n = 20
S₂₀ = 20/2 [50 + 38]
= 10 × 88 = 880
Total seats = 880
Exam-Oriented Shortcuts
✔ Use nth term formula for position-based questions
✔ Use sum formula when total is asked
✔ Convert word problems into AP form carefully
Common Mistakes in AP
- Confusing n with last term
- Incorrect substitution in formula
- Forgetting brackets in (n − 1)d
- Calculation mistakes in large sums
Previous Year Exam Pattern Insight
In CBSE board exams, Arithmetic Progression generally carries 4–6 marks. Questions are usually based on:
- Finding nth term
- Finding sum of n terms
- Application-based word problems
Final Revision Strategy
Practice at least 15 mixed problems covering:
- Direct nth term
- Sum calculation
- Finding missing term
- Word problems
Mastering AP ensures you secure easy marks in board exams.