class 11 notes

Class 11 Mathematics Notes Chapter 9 (Sequences and series) – Mathematics Book

Philoid

Philoid

— 6 min read

Mathematics
Alright class, let's get focused on Chapter 9: Sequences and Series. This is a fundamental chapter, and concepts from here frequently appear in various government exams, often testing your understanding of patterns and summations. Pay close attention to the definitions, formulas, and their applications.

Chapter 9: Sequences and Series - Detailed Notes for Government Exam Preparation

1. Sequence:

  • Definition: A sequence is an ordered list of numbers (called terms) arranged according to a specific rule.
  • Notation: Terms are denoted by a₁, a₂, a₃, ..., aₙ, ... where aₙ is the nth term.
  • Finite Sequence: A sequence with a limited number of terms.
  • Infinite Sequence: A sequence that continues indefinitely.
  • Finding Terms: Often, the nth term (general term) aₙ is given by a formula involving 'n'. You can find specific terms by substituting n = 1, 2, 3, ...

Example: If aₙ = 2n + 1, the sequence is a₁ = 2(1)+1=3, a₂ = 2(2)+1=5, a₃ = 2(3)+1=7, ... i.e., 3, 5, 7, ...

2. Series:

  • Definition: A series is the sum of the terms of a sequence.
  • Notation: If a₁, a₂, a₃, ... is a sequence, the corresponding series is a₁ + a₂ + a₃ + ... . The sum of the first n terms is denoted by Sₙ.
  • Sigma Notation (Σ): Used to represent series compactly.
    • Sₙ = a₁ + a₂ + ... + aₙ = Σ_{k=1}^{n} aₖ

3. Arithmetic Progression (AP):

  • Definition: A sequence where the difference between any term and its preceding term is constant. This constant difference is called the common difference (d).
    • d = aₙ₊₁ - aₙ
  • General Term (nth term): If the first term is 'a' and the common difference is 'd', then:
    • aₙ = a + (n-1)d
  • Sum of First n Terms (Sₙ):
    • Sₙ = n/2 [2a + (n-1)d]
    • Sₙ = n/2 [a + l] (where l = aₙ is the last term)
  • Arithmetic Mean (AM):
    • The AM of two numbers 'a' and 'b' is (a+b)/2.
    • If a, A, b are in AP, then A = (a+b)/2 is the single AM between a and b.
    • To insert 'n' AMs between 'a' and 'b', we create an AP with (n+2) terms where the first term is 'a' and the last term is 'b'. The common difference is d = (b-a)/(n+1). The inserted means are a+d, a+2d, ..., a+nd.
  • Properties of AP:
    • If a constant is added/subtracted to each term, the resulting sequence is also an AP with the same common difference.
    • If each term is multiplied/divided by a non-zero constant 'k', the resulting sequence is also an AP with common difference kd or d/k respectively.
    • In a finite AP, the sum of terms equidistant from the beginning and end is constant and equal to the sum of the first and last terms (a₁ + aₙ = a₂ + aₙ₋₁ = ...).
    • Three terms in AP: a-d, a, a+d.
    • Four terms in AP: a-3d, a-d, a+d, a+3d.
    • Five terms in AP: a-2d, a-d, a, a+d, a+2d.

4. Geometric Progression (GP):

  • Definition: A sequence where the ratio of any term to its preceding term is constant (and non-zero). This constant ratio is called the common ratio (r).
    • r = aₙ₊₁ / aₙ
  • General Term (nth term): If the first term is 'a' and the common ratio is 'r', then:
    • aₙ = arⁿ⁻¹
  • Sum of First n Terms (Sₙ):
    • Sₙ = a(rⁿ - 1) / (r - 1) or Sₙ = a(1 - rⁿ) / (1 - r), provided r ≠ 1.
    • If r = 1, Sₙ = na.
  • Sum of Infinite Geometric Series (S<0xE2><0x88><0x9E>):
    • If the common ratio 'r' satisfies |r| < 1 (i.e., -1 < r < 1), the sum of an infinite GP converges to:
    • S<0xE2><0x88><0x9E> = a / (1 - r)
    • If |r| ≥ 1, the sum diverges (does not approach a finite value).
  • Geometric Mean (GM):
    • The GM of two positive numbers 'a' and 'b' is √(ab).
    • If a, G, b are in GP, then G = √(ab) is the single GM between a and b (assuming a, b have the same sign).
    • To insert 'n' GMs between 'a' and 'b', we create a GP with (n+2) terms. The common ratio is r = (b/a)^(1/(n+1)). The inserted means are ar, ar², ..., arⁿ.
  • Properties of GP:
    • If each term is multiplied/divided by a non-zero constant, the resulting sequence is also a GP with the same common ratio.
    • The reciprocals of the terms of a GP also form a GP.
    • If a₁, a₂, ..., aₙ are in GP, then log a₁, log a₂, ..., log aₙ are in AP (base of log > 0, ≠ 1).
    • Three terms in GP: a/r, a, ar.
    • Four terms in GP: a/r³, a/r, ar, ar³.
    • Five terms in GP: a/r², a/r, a, ar, ar².

5. Relationship Between Arithmetic Mean (AM) and Geometric Mean (GM):

  • For any two positive numbers 'a' and 'b':
    • AM ≥ GM
    • (a+b)/2 ≥ √(ab)
  • Equality holds (AM = GM) if and only if a = b.
  • This relationship is crucial for minimum/maximum value problems.

6. Sum of First n Terms of Special Series:

  • Sum of first n natural numbers:
    • Σ_{k=1}^{n} k = 1 + 2 + 3 + ... + n = n(n+1)/2
  • Sum of squares of first n natural numbers:
    • Σ_{k=1}^{n} k² = 1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6
  • Sum of cubes of first n natural numbers:
    • Σ_{k=1}^{n} k³ = 1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]² = (Σk)²

Exam Focus Points:

  • Memorize the formulas for aₙ, Sₙ (AP & GP), S<0xE2><0x88><0x9E> (GP), and the special series sums.
  • Understand the conditions under which formulas apply (e.g., |r|<1 for S<0xE2><0x88><0x9E>).
  • Be able to identify whether a sequence is an AP, GP, or neither.
  • Practice problems involving finding terms, sums, inserting means, and applying properties.
  • The AM ≥ GM inequality is a favorite in competitive exams.

Multiple Choice Questions (MCQs):

  1. The 10th term of the AP: 5, 8, 11, 14, ... is:
    (a) 32
    (b) 35
    (c) 38
    (d) 29

  2. The sum of the first 16 terms of the AP: 10, 6, 2, ... is:
    (a) -320
    (b) 320
    (c) -352
    (d) -400

  3. What is the single Arithmetic Mean (AM) between 4 and 16?
    (a) 8
    (b) 10
    (c) 12
    (d) 6

  4. The 6th term of the GP: 2, 8, 32, ... is:
    (a) 1024
    (b) 2048
    (c) 512
    (d) 4096

  5. The sum of the first 5 terms of the GP: 3, 6, 12, ... is:
    (a) 93
    (b) 90
    (c) 81
    (d) 96

  6. The sum of the infinite GP: 1, 1/3, 1/9, 1/27, ... is:
    (a) 3/2
    (b) 2/3
    (c) 1
    (d) ∞

  7. If A and G are the Arithmetic Mean and Geometric Mean between two positive numbers respectively, then:
    (a) A < G
    (b) A > G
    (c) A ≤ G
    (d) A ≥ G

  8. The value of 1² + 2² + 3² + ... + 10² is:
    (a) 385
    (b) 55
    (c) 3025
    (d) 400

  9. If the 3rd term of an AP is 12 and the 7th term is 24, what is the 10th term?
    (a) 30
    (b) 33
    (c) 36
    (d) 39

  10. In a GP, the first term is 5 and the common ratio is 2. Which term of the GP is 160?
    (a) 4th
    (b) 5th
    (c) 6th
    (d) 7th

Answer Key:

  1. (a) 32 [a=5, d=3. a₁₀ = a + 9d = 5 + 9(3) = 5 + 27 = 32]
  2. (a) -320 [a=10, d=-4, n=16. S₁₆ = 16/2 [2(10) + (16-1)(-4)] = 8 [20 - 60] = 8[-40] = -320]
  3. (b) 10 [AM = (4+16)/2 = 20/2 = 10]
  4. (b) 2048 [a=2, r=4. a₆ = ar⁵ = 2 * (4)⁵ = 2 * 1024 = 2048]
  5. (a) 93 [a=3, r=2, n=5. S₅ = a(rⁿ - 1)/(r - 1) = 3(2⁵ - 1)/(2 - 1) = 3(32 - 1)/1 = 3 * 31 = 93]
  6. (a) 3/2 [a=1, r=1/3. |r|<1. S<0xE2><0x88><0x9E> = a/(1-r) = 1 / (1 - 1/3) = 1 / (2/3) = 3/2]
  7. (d) A ≥ G [Standard result for positive numbers]
  8. (a) 385 [Sum of squares = n(n+1)(2n+1)/6. For n=10, 10(11)(21)/6 = 5(11)(7) = 385]
  9. (b) 33 [a₃ = a+2d = 12; a₇ = a+6d = 24. Subtracting: 4d = 12 => d=3. Substituting d=3 in a+2d=12 => a+6=12 => a=6. a₁₀ = a+9d = 6 + 9(3) = 6 + 27 = 33]
  10. (c) 6th [a=5, r=2. Let aₙ = 160. arⁿ⁻¹ = 160 => 5 * 2ⁿ⁻¹ = 160 => 2ⁿ⁻¹ = 32 => 2ⁿ⁻¹ = 2⁵. So, n-1 = 5 => n=6]

Make sure you understand the reasoning behind each answer. Practice similar problems to solidify these concepts. Good luck!

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