Last Updated: May 2026
CBSE Class 10 Maths Chapter 5 — Arithmetic Progressions (AP) is one of the most scoring chapters in the CBSE Class 10 board exam. Typically worth 5–8 marks across MCQs and short-answer questions, AP rewards students who memorise just three formulas and practice 30 carefully-chosen problems. This guide gives complete NCERT solutions, formulas, common pitfalls, and 25 practice MCQs.
Chapter Overview
- NCERT Pages: 5.1 to 5.4 (~22 pages)
- Board exam weightage: 5–8 marks
- Topics: AP definition, n-th term, sum of n terms, applications
1. What is an Arithmetic Progression?
A sequence of numbers where each term differs from the previous one by a fixed quantity called the common difference (d).
General form: a, a+d, a+2d, a+3d, …, a+(n−1)d
Where:
- a = first term
- d = common difference (can be positive, negative, or zero)
- n = number of terms
Examples
- 2, 5, 8, 11, … → a = 2, d = 3
- 10, 7, 4, 1, … → a = 10, d = −3
- −5, −2, 1, 4, … → a = −5, d = 3
2. Three Key Formulas (Memorise These)
| Formula | Use |
|---|---|
| an = a + (n−1)d | Find the n-th term of an AP |
| Sn = (n/2)[2a + (n−1)d] | Sum of first n terms |
| Sn = (n/2)(a + l) | Sum when first term a and last term l are known |
3. Worked Examples (Board Pattern)
Example 1 (2 marks) — Find the 15th term
The AP is 7, 13, 19, … Find the 15th term.
Solution: a = 7, d = 6, n = 15. a₁₅ = 7 + (15−1)×6 = 7 + 84 = 91.
Example 2 (3 marks) — Find n given an
Which term of the AP 3, 8, 13, … is 78?
Solution: 78 = 3 + (n−1)×5 → 75 = (n−1)×5 → n−1 = 15 → n = 16. So 78 is the 16th term.
Example 3 (3 marks) — Find Sn
Find the sum of first 25 terms of the AP 5, 8, 11, …
Solution: a = 5, d = 3, n = 25. S₂₅ = (25/2)[2×5 + 24×3] = (25/2)[10 + 72] = (25/2)×82 = 1025.
Example 4 (4 marks) — Word Problem
A man saves ₹100 in the first month, ₹150 in the second, ₹200 in the third, and so on. How much will he save in 12 months?
Solution: a = 100, d = 50, n = 12. S₁₂ = (12/2)[200 + 11×50] = 6 × 750 = ₹4500.
Example 5 (4 marks) — Find AP from Conditions
If the 7th term of an AP is 40 and 13th term is 64, find the AP.
Solution: a + 6d = 40, a + 12d = 64. Subtracting: 6d = 24 → d = 4. So a = 40 − 24 = 16. AP: 16, 20, 24, 28, …
4. Important Properties of AP
- If a, b, c are in AP → 2b = a + c (b is the arithmetic mean of a and c)
- If you add (or subtract) the same number to each term, the result is still an AP with the same d
- If you multiply each term by the same non-zero number, the result is an AP with new d
- The middle term of an AP with odd number of terms is the average of all terms
- If Sn denotes sum, then an = Sn − Sn−1
5. Common Mistakes to Avoid
- Forgetting (n−1) in the n-th term formula — using “nd” instead of “(n−1)d”
- Wrong sign of d when AP is decreasing
- Counting terms wrongly — students confuse the count of terms vs the value of n-th term
- Mixing up Sn formulas — use (n/2)(a+l) only when last term is given
- Errors in algebra when solving simultaneous equations for a and d
6. Quick-Reference Identities
| Series | Sum |
|---|---|
| 1 + 2 + 3 + … + n | n(n+1)/2 |
| 2 + 4 + 6 + … + 2n (even) | n(n+1) |
| 1 + 3 + 5 + … + (2n−1) (odd) | n² |
7. Strategy for Board Exam
- Memorise all three formulas in 5 minutes — they don’t change
- Always identify a, d, n at the start of each problem
- For “which term is X” questions — set an = X and solve for n
- For sum problems — check whether last term is given (use shorter formula) or not
- Word problems: convert to a, d, n in the first 30 seconds
NCERT Exercises Coverage
- Exercise 5.1: 4 problems — basic identification of AP
- Exercise 5.2: 20 problems — finding n-th term, given conditions
- Exercise 5.3: 20 problems — sum of n terms, applications
- Exercise 5.4 (Optional): 5 problems — advanced applications
25 Practice MCQs
Quiz data missing.
FAQ
Q1. How many marks does Arithmetic Progressions carry?
5–8 marks combined — typically 1 MCQ (1 mark) + 1 short answer (2-3 marks) + 1 long answer (4 marks).
Q2. Is the AP chapter difficult?
No — it’s one of the easiest scoring chapters once you memorise three formulas. Practice 30 problems and you’ll get every board question.
Q3. Are word problems common?
Yes — at least one word problem (savings, salary increment, distance, etc.) appears every year.
Q4. What if I forget the n-th term formula?
Derive it from the pattern: a1=a, a2=a+d, a3=a+2d, …, an=a+(n−1)d. Visualise the pattern.
Q5. Are AP questions repeated from sample papers?
Direct repetition is rare, but the structure of word problems (savings/distance/salary) repeats — practice CBSE sample papers from 2020–2026.
