Sequences and Series – Class 11 Maths Chapter 8 Notes & NCERT PDF

Class 11MathematicsChapter 8NCERT book: Mathematics - Class 11

Notes and a simple summary of Chapter 8 of the NCERT Class 11 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 8: Sequences and Series

Introduction

A sequence is an ordered collection of objects with a first, second, third member and so on — for example, populations at different times, or money in a bank over the years. Sequences that follow a specific pattern are called progressions. Having met the arithmetic progression (A.P.) in an earlier class, this chapter (rationalised syllabus) studies sequences and series in general, the geometric progression (G.P.), the geometric mean (G.M.) and the relationship between A.M. and G.M.

Key Concepts

8.2 Sequences

The terms of a sequence are written a₁, a₂, a₃, …, aₙ, …; aₙ is the nth term (general term).

Finite sequence: a fixed number of terms (e.g. ancestors over 10 generations: 2, 4, 8, …, 1024 — with a 30-year generation gap, 300 years = 10 generations).

Infinite sequence: not finite (e.g. the successive quotients 3, 3.3, 3.33, … when dividing 10 by 3).

Terms may come from a formula (even numbers aₙ = 2n, odd numbers aₙ = 2n − 1), from a recurrence relation, or only from a description (the primes 2, 3, 5, 7, … have no formula for the nth term).

Fibonacci sequence: a₁ = a₂ = 1, aₙ = aₙ₋₂ + aₙ₋₁ for n > 2 → 1, 1, 2, 3, 5, 8, …

A sequence can be regarded as a function whose domain is the natural numbers (or a subset {1, 2, …, k}).

8.3 Series

For a sequence a₁, a₂, …, aₙ, the expression a₁ + a₂ + … + aₙ is the associated series, written Σ aₖ (k = 1 to n). A series refers to the indicated sum; the "sum of the series" is the resulting number (1 + 3 + 5 + 7 has sum 16).

8.4 Geometric Progression

A sequence is a G.P. if each term (except the first) bears a constant ratio to the term before it: aₖ₊₁/aₖ = r. With first term a and common ratio r, the G.P. is a, ar, ar², ar³, …

General term: aₙ = arⁿ⁻¹.

Sum of first n terms: Sₙ = a(rⁿ − 1)/(r − 1) or a(1 − rⁿ)/(1 − r) for r ≠ 1; Sₙ = na if r = 1.

Geometric mean of two positive numbers a and b: G = √(ab) (G.M. of 2 and 8 is 4). To insert n G.M.s between a and b, use b = arⁿ⁺¹, so r = (b/a)^(1/(n+1)).

8.5 Relationship between A.M. and G.M.

For positive real numbers a and b, A = (a + b)/2 and G = √(ab). Then A − G = (√a − √b)²/2 ≥ 0, so A ≥ G (equality only when a = b).

Worked Examples (from the chapter)

Example 1: aₙ = 2n + 5 → first three terms 7, 9, 11.

Example 2: aₙ = (n − 1)(2 − n)(3 + n) ⇒ a₂₀ = 19 × (−18) × 23 = −7866.

Example 3: a₁ = 1, aₙ = aₙ₋₁ + 2 → 1, 3, 5, 7, 9; series 1 + 3 + 5 + 7 + 9.

Example 4: G.P. 5, 25, 125, … : a₁₀ = 5·5⁹ = 5¹⁰, aₙ = 5ⁿ.

Example 5: Which term of 2, 8, 32, … is 131072? 2·4ⁿ⁻¹ = 131072 ⇒ 4ⁿ⁻¹ = 65536 = 4⁸ ⇒ n = 9.

Example 6: 3rd term 24, 6th term 192 ⇒ r³ = 8, r = 2, a = 6 ⇒ a₁₀ = 6·2⁹ = 3072.

Example 7: G.P. 1, 2/3, 4/9, … : Sₙ = 3[1 − (2/3)ⁿ]; S₅ = 3(1 − 32/243) = 211/81.

Example 8: Terms of 3, 3/2, 3/4, … needed for sum 3069/512: 6(1 − 1/2ⁿ) = 3069/512 ⇒ 2ⁿ = 1024 ⇒ n = 10.

Example 9: Sum of three G.P. terms 13/12, product −1. Take a/r, a, ar: a³ = −1 ⇒ a = −1; then 12r² + 25r + 12 = 0 ⇒ r = −3/4 or −4/3. Terms: 4/3, −1, 3/4 (or the reverse order).

Example 10: 7 + 77 + 777 + … to n terms = (7/9)[(10 + 100 + … ) − n] = (70/81)(10ⁿ − 1) − 7n/9.

Example 11: Ancestors over 10 generations: 2 + 4 + … = 2(2¹⁰ − 1) = 2046.

Example 12: Insert three numbers between 1 and 256 in G.P.: r⁴ = 256 ⇒ r = 4 → 4, 16, 64 (or −4, 16, −64 for r = −4).

Example 13: A.M. = 10, G.M. = 8 ⇒ a + b = 20, ab = 64 ⇒ (a − b)² = 144 ⇒ numbers 16 and 4.

Example 14 (miscellaneous): If (a² + b² + c²)p² − 2(ab + bc + cd)p + (b² + c² + d²) ≤ 0, the left side equals (ap − b)² + (bp − c)² + (cp − d)², which must be 0, so b/a = c/b = d/c = p: a, b, c, d are in G.P.

Formula Sheet

aₙ = arⁿ⁻¹

Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1; Sₙ = na, r = 1

G.M. of a, b: √(ab)

n G.M.s between a and b: r = (b/a)^(1/(n+1))

A.M. ≥ G.M. for positive a, b

Three terms in G.P.: a/r, a, ar (product a³)

Exam Focus

Identify a and r first; for "which term" problems equate arⁿ⁻¹ to the given value and compare powers.

For "sum of first three terms and product" problems, always take a/r, a, ar.

For 7 + 77 + 777, 8 + 88 + 888 … write each term as (7/9)(10ᵏ − 1).

Growth problems (bacteria doubling, compound interest) are G.P. questions: 30 bacteria doubling each hour give 120 after 2 hours and 480 after 4 hours.

Answer outline — "The 5th, 8th and 11th terms of a G.P. are p, q, s. Show q² = ps": p = ar⁴, q = ar⁷, s = ar¹⁰ ⇒ ps = a²r¹⁴ = (ar⁷)² = q².

Key Terms

Sequence (अनुक्रम)

Series (श्रेणी)

Term / general term (पद / व्यापक पद)

Finite / infinite sequence (परिमित / अपरिमित अनुक्रम)

Progression (श्रेढ़ी)

Geometric progression (गुणोत्तर श्रेढ़ी)

Common ratio (सार्व अनुपात)

Geometric mean (गुणोत्तर माध्य)

Arithmetic mean (समांतर माध्य)

Recurrence relation (पुनरावृत्ति संबंध)

Fibonacci sequence (फिबोनाची अनुक्रम)

Common Mistakes

Writing aₙ = arⁿ instead of arⁿ⁻¹.

Using Sₙ = a(rⁿ − 1)/(r − 1) when r = 1 (then Sₙ = na).

Taking only r = 4 in Example 12 and ignoring r = −4.

Treating 7 + 77 + 777 as a G.P. directly — it is not.

Forgetting that the G.M. is defined here for positive numbers, and that A ≥ G needs positivity.

Calling a series "the sum" — a series is the indicated sum.

💡 Key Learning Points

✓Define sequences (finite and infinite), general terms, recurrence relations such as Fibonacci, and the associated series.
✓Recognise a geometric progression and use aₙ = arⁿ⁻¹ to find any term.
✓Find the sum of n terms of a G.P. using Sₙ = a(rⁿ − 1)/(r − 1), and Sₙ = na when r = 1.
✓Find and insert geometric means, taking three terms as a/r, a, ar where convenient.
✓Prove and apply A.M. ≥ G.M. for positive numbers.

👨‍🏫 Teaching Tips

→Start with the book's ancestor example (2, 4, 8, …, 1024 over 10 generations) and ask students to guess the total before computing 2046.
→Contrast A.P. and G.P. side by side: adding a fixed number vs multiplying by a fixed number.
→Derive the G.P. sum formula live by writing Sₙ and rSₙ and subtracting.
→Use a bacteria-doubling or compound-interest story to show G.P. growth in real life.
→Model 7 + 77 + 777 … by rewriting each term as 7/9 × (10ᵏ − 1).
→Illustrate A ≥ G numerically with pairs such as (2, 8) and (4, 16) before the algebraic proof.
→Let students generate the Fibonacci sequence and spot it in nature pictures.

📋 Assessment Questions

Can the student find a specified term of a G.P. and identify which term equals a given value?
Can the student compute the sum of n terms of a G.P., including fractional ratios?
Can the student insert geometric means between two numbers?
Can the student solve three-term G.P. problems using a/r, a, ar?
Can the student prove A.M. ≥ G.M. and use it to find two numbers from their A.M. and G.M.?

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