Arithmetic Progressions — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Arithmetic Progressions (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Arithmetic Progressions (CBSE & HBSE)
Arithmetic Progressions is a formulaic chapter with direct application. CBSE tests finding nth term and sum of n terms, including word problems. HBSE focuses on identifying APs and applying formulas.
Introduction to AP and nth Term
What is an Arithmetic Progression?
Definition: A sequence where the difference between consecutive terms is constant.
This constant difference is called the Common Difference (d).
General AP: a, a+d, a+2d, a+3d, ...
- First term = a
- Common difference = d (can be positive, negative, or zero)
Identifying an AP
For sequence a₁, a₂, a₃, ... to be an AP: a₂ - a₁ = a₃ - a₂ = a₄ - a₃ = ... = d (constant)
| Sequence | d | AP? |
|---|---|---|
| 2, 5, 8, 11 | 3 | Yes |
| 1, 1, 1, 1 | 0 | Yes (constant AP) |
| 1, 4, 9, 16 | Varies (3, 5, 7...) | No |
| 10, 7, 4, 1 | -3 | Yes (decreasing AP) |
nth Term Formula
$$a_n = a + (n-1)d$$
Where:
- aₙ = nth term
- a = first term (a₁)
- d = common difference
- n = position (n ≥ 1)
Example: AP: 3, 7, 11, 15, ... Find 20th term.
- a = 3, d = 4
- a₂₀ = 3 + 19×4 = 3 + 76 = 79
Finding the Position of a Term
If aₙ = k, solve: a + (n-1)d = k for n.
Example: Which term of AP 5, 8, 11, 14, ... is 89?
- 5 + (n-1)×3 = 89
- (n-1) = 28 → n = 29 (29th term)
Exam Tip: If asked whether a given value is a term, solve for n. If n is a positive integer, it's a term; otherwise it isn't.
Sum of n Terms of an AP
Sum Formula
$$S_n = \frac{n}{2}[2a + (n-1)d]$$
Alternative form (when last term l = aₙ is known): $$S_n = \frac{n}{2}[a + l]$$
Where l = a + (n-1)d = last term
Derivation Intuition
Sum = (first + last)/2 × number of terms = average × count
Key Relationships
- aₙ = Sₙ - Sₙ₋₁ (nth term = difference of consecutive sums)
- If Sₙ = pn² + qn, then: a = p + q, d = 2p
Worked Example
Find sum of first 50 terms of AP: 1, 3, 5, 7, ... (odd numbers)
- a = 1, d = 2, n = 50
- S₅₀ = 50/2 × [2(1) + 49(2)] = 25 × [2 + 98] = 25 × 100 = 2500
Quick check: Sum of first n odd numbers = n²: 50² = 2500 ✓
Sum of First n Natural Numbers
- S = n(n+1)/2 (d=1, a=1)
- This is a special case of the AP sum formula
When to use Sₙ vs aₙ
| Question asks | Use |
|---|---|
| Value of a specific term | aₙ = a+(n-1)d |
| Sum of terms | Sₙ = n/2[2a+(n-1)d] |
| Which term equals k | Solve aₙ = k for n |
| How many terms needed for sum = k | Solve Sₙ = k for n |
HBSE Favourite: If Sₙ = an² + bn, find a, d and the AP.
Applications and Word Problems
Word Problem Strategy for AP
- Identify the progression — is it increasing/decreasing by a constant amount?
- Write a = first value, d = constant change, n = number of terms
- Choose formula: aₙ or Sₙ based on what's asked
- Solve and verify your answer
Common AP Word Problem Templates
Template 1: Savings/Deposits "Suman deposits Rs 200 in first month, Rs 250 in second, Rs 300 in third... How much in 12th month? Total in a year?"
- a = 200, d = 50
- a₁₂ = 200 + 11×50 = Rs 750 (12th month)
- S₁₂ = 12/2 × (200 + 750) = 6 × 950 = Rs 5700 (total)
Template 2: Stadium/Seating Rows "Seats in rows: 23, 27, 31, ... Row 1 has 23 seats, d = 4."
Template 3: Fallen Objects / Free Fall Distances fallen in consecutive seconds: 5, 15, 25, 35, ... (d = 10)
Template 4: Penalties/Fines "A person pays Rs 200 as fine, Rs 250 next month, increasing Rs 50 each month."
Three Consecutive Terms in AP
If three terms are in AP, take them as: a-d, a, a+d (middle term is average)
Advantage: Sum of three terms = 3a (simpler equations!)
If four consecutive AP terms are a-3d, a-d, a+d, a+3d (sum = 4a)
Example: Three terms in AP have sum 24 and product 480.
- Sum: (a-d) + a + (a+d) = 3a = 24 → a = 8
- Product: (8-d)(8)(8+d) = 480 → 8(64-d²) = 480 → d² = 4 → d = ±2
- Terms: 6, 8, 10 or 10, 8, 6
Key Trick: Assume AP terms as a-d, a, a+d for 3 terms or a-3d, a-d, a+d, a+3d for 4 terms to simplify sum calculations.
Frequently asked questions
Are these Arithmetic Progressions notes free?
Yes — the Arithmetic Progressions notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Arithmetic Progressions notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Arithmetic Progressions chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Arithmetic Progressions.