Arithmetic Progressions Class 10 Notes: nth Term, Sum Formula and Solved Examples

Arithmetic Progressions (Chapter 5) is worth 5–6 marks and is one of the easiest chapters to score full marks in, because it uses only two formulas.

Definition

A list where each term is obtained by adding a fixed number d (common difference) to the previous term. First term a. Example: 3, 7, 11, 15 (a = 3, d = 4).

Formulas

• nth term: aₙ = a + (n − 1)d
• Sum of n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term
• nth term from the end: l − (n − 1)d
• aₙ = Sₙ − Sₙ₋₁
• Three terms in AP: a − d, a, a + d; four terms: a − 3d, a − d, a + d, a + 3d

Solved examples

1. Which term of 5, 8, 11, … is 320? 320 = 5 + (n − 1)3 → n − 1 = 105 → n = 106.

2. Sum of the first 20 terms of 2, 7, 12, …: S₂₀ = 20/2 [4 + 19 × 5] = 10 × 99 = 990.

3. How many two-digit numbers are divisible by 3? 12, 15, …, 99: 99 = 12 + (n − 1)3 → n = 30.

4. If the 3rd term is 16 and the 7th exceeds the 5th by 12: a + 2d = 16, 2d = 12 → d = 6, a = 4 → AP: 4, 10, 16, …

5. Three numbers in AP have sum 24 and product 440: 3a = 24 → a = 8; (8 − d)8(8 + d) = 440 → 64 − d² = 55 → d = ±3 → 5, 8, 11.

Common mistakes

• Using n instead of n − 1 in the nth term
• Forgetting that d can be negative or a fraction
• Using Sₙ = n/2 (a + l) when l is not the nth term

Related: Quadratic Equations Class 10 notes.

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Frequently asked questions

Are these Arithmetic Progressions Class 10 Notes notes enough for the exam?

They cover every formula, definition and question type that appears in school and board papers for this chapter. Use them with the NCERT examples, exercise questions and the last five years' papers to be fully prepared.

How should I revise Arithmetic Progressions Class 10 Notes?

Revise the notes after 1 day, 7 days and 30 days. Solve five questions each time instead of only reading, and keep an error list of the steps you get wrong.

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