Chapter 2 of 8

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Tamil Nadu (TNBSE)Class 10 Mathematics★ 8-12 marks in the TN SSLC Class 10 Maths public exam through MCQs, short answers and AP/GP numericals

Numbers and Sequences

Samacheer Kalvi (TN State Board) Class 10 Maths Chapter 2, Numbers and Sequences. Euclid's division lemma and algorithm, the fundamental theorem of arithmetic, modular arithmetic, arithmetic and geometric progressions with their sums, and special series. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 22 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Apply Euclid's division lemma and algorithm to find HCF
2State the fundamental theorem of arithmetic
3Find the nth term and sum of an AP
4Find the nth term and sum of a GP
5Use the special-series formulas

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Why this chapter matters

Numbers and Sequences combines number theory with progressions. Euclid's algorithm, AP and GP formulas and special series are heavily tested and reliably scoring in the TN SSLC exam.

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A 5-minute refresher here will save you 30 minutes of confusion below.

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Class 9 — Real numbers

Integers, primes and divisibility.

Numbers and Sequences — Class 10 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 10 Mathematics, Chapter 2. Number theory basics and the arithmetic of sequences.

1. About this chapter

This chapter covers Euclid's division lemma and algorithm, the fundamental theorem of arithmetic, modular arithmetic, arithmetic (AP) and geometric (GP) progressions, and special series.

2. Number theory

Euclid's division lemma: for positive integers a, b there exist unique q, r with a = bq + r, 0 ≤ r < b.
Euclid's division algorithm: repeated use of the lemma gives the HCF of two numbers.
Fundamental theorem of arithmetic: every composite number can be written as a unique product of primes (apart from order).
Modular arithmetic: working with remainders (a ≡ b (mod n)).

3. Arithmetic progression (AP)

General form: a, a + d, a + 2d, … with common difference d.
nth term: tₙ = a + (n − 1)d.
Sum of n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l) (l = last term).

4. Geometric progression (GP) and special series

General form: a, ar, ar², … with common ratio r.
nth term: tₙ = a r^(n−1).
Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) (r ≠ 1).
Special series: Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².

5. Worked examples

Example 1. Find the HCF of 24 and 36 using Euclid's algorithm. 36 = 24×1 + 12; 24 = 12×2 + 0 → HCF = 12.

Example 2. Find the 10th term of the AP 3, 7, 11, … a = 3, d = 4; t₁₀ = a + 9d = 3 + 36 = 39.

Example 3. Find the sum of the first 20 natural numbers. Σn = n(n+1)/2 = 20×21/2 = 210.

Example 4. Find the sum of the GP 2 + 6 + 18 + … to 5 terms. a = 2, r = 3; S₅ = 2(3⁵ − 1)/(3 − 1) = 2(243 − 1)/2 = 242.

6. Common mistakes

Mistake: Using r = 1 in the GP sum formula. Fix: If r = 1, Sₙ = na (the formula needs r ≠ 1).
Mistake: Forgetting (n − 1) in the nth-term formulas. Fix: AP tₙ = a + (n − 1)d; GP tₙ = a r^(n − 1).
Mistake: Confusing common difference and common ratio. Fix: AP has a common difference; GP has a common ratio.

7. Practice (book-back style)

State Euclid's division lemma.
Find the 15th term of the AP 5, 8, 11, …
Find the sum of the first 12 terms of the AP 2, 5, 8, …
Find the 6th term of the GP 3, 6, 12, …
Find Σn² for n = 10.

8. Answer key

a = bq + r with 0 ≤ r < b, for positive integers a and b.
a = 5, d = 3; t₁₅ = 5 + 14×3 = 47.
S₁₂ = 12/2[2(2) + 11(3)] = 6[4 + 33] = 6×37 = 222.
a = 3, r = 2; t₆ = 3×2⁵ = 96.
Σn² = 10×11×21/6 = 385.

9. Quick revision

Chapter 2 · number theory + AP/GP + special series.
Euclid: a = bq + r; algorithm → HCF; FTA → unique prime factorisation.
AP: tₙ = a + (n−1)d; Sₙ = n/2[2a + (n−1)d].
GP: tₙ = a r^(n−1); Sₙ = a(rⁿ − 1)/(r − 1).
Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Euclid's division lemma

a = bq + r, 0 ≤ r < b

Basis of the HCF algorithm.

AP nth term / sum

tₙ = a + (n−1)d ; Sₙ = n/2[2a + (n−1)d]

Also Sₙ = n/2 (a + l).

GP nth term / sum

tₙ = a r^(n−1) ; Sₙ = a(rⁿ − 1)/(r − 1)

For r ≠ 1.

Special series

Σn = n(n+1)/2 ; Σn² = n(n+1)(2n+1)/6 ; Σn³ = [n(n+1)/2]²

Sums of natural numbers, squares, cubes.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Using r = 1 in the GP sum formula

✓ If r = 1 the sum is Sₙ = na; the formula needs r ≠ 1.

WATCH OUT

✗ Forgetting (n − 1) in nth-term formulas

✓ AP tₙ = a + (n−1)d; GP tₙ = a r^(n−1).

WATCH OUT

✗ Confusing common difference and common ratio

✓ AP has a common difference; GP has a common ratio.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Multiple choice questions

MCQs on number theory and progressions.

Questions

Short answer questions

Euclid's algorithm, nth term and series.

Questions

AP/GP numericals

Term and sum problems for AP, GP and special series.

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Concept

State Euclid's division lemma.

▶ Show solution

For positive integers a and b there exist unique q, r with a = bq + r and 0 ≤ r < b.

Q2EASY· Numerical

Find the 15th term of the AP 5, 8, 11, …

▶ Show solution

a = 5, d = 3; t₁₅ = 5 + 14×3 = 47.

Q3MEDIUM· Numerical

Find the sum of the first 12 terms of the AP 2, 5, 8, …

▶ Show solution

S₁₂ = 12/2[2(2)+11(3)] = 6(4+33) = 222.

Q4MEDIUM· Numerical

Find the 6th term of the GP 3, 6, 12, …

▶ Show solution

a = 3, r = 2; t₆ = 3×2⁵ = 96.

Q5MEDIUM· Numerical

Find Σn² for n = 10.

▶ Show solution

Σn² = 10×11×21/6 = 385.

Q6EASY· Numerical

Find the HCF of 24 and 36 using Euclid's algorithm.

▶ Show solution

36 = 24×1 + 12; 24 = 12×2 + 0 → HCF = 12.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Chapter 2 of Samacheer Kalvi Class 10 Mathematics.
•Euclid: a = bq + r; algorithm → HCF; FTA → unique prime factorisation.
•AP: tₙ = a + (n−1)d; Sₙ = n/2[2a + (n−1)d].
•GP: tₙ = a r^(n−1); Sₙ = a(rⁿ − 1)/(r − 1).
•Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².
•Check r ≠ 1 before using the GP sum formula.

Tamil Nadu (TNBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 8-12 marks across MCQ, short answer and numericals

Question type	Marks each	Typical count	What it tests
MCQ	1	1-2	Number theory and progressions
Short Answer	2-3	2-3	Euclid, nth term, special series
Numerical	2-5	1-2	AP/GP term and sum

Prep strategy

Memorise AP and GP term/sum formulas
Practise Euclid's algorithm for HCF
Learn the three special-series formulas
Watch r ≠ 1 in the GP sum

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Finance

GP models compound interest and population growth.

Scheduling

AP describes evenly spaced events like instalments.

Cryptography

Modular arithmetic underlies coding and security.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Write the formula before substituting
Identify a, d (AP) or a, r (GP) first
Use special-series formulas for sums of powers
Show each step in Euclid's algorithm

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Prove the sum of the first n odd numbers is n².
Find three numbers in GP given their sum and product.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

TN SSLC Class 10 Public ExamHigh

Foundation / NTSE MathematicsMedium

School unit testsHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Apply the division lemma repeatedly — divide, take the remainder, and divide again — until the remainder is zero; the last non-zero divisor is the HCF.

When the common difference d = 0, every term equals the first term, giving a constant sequence.

Practice more, ask doubts, find a tutor

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