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Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 1 Chapter 4: Sequences and Series

Short notes, key concepts, quick revision and practice MCQs with answers

A sequence is a list of numbers in a definite order. After recalling the arithmetic progression, the chapter studies the geometric progression (G.P.): its nth term, the sum of n terms and the sum to infinity, then the relation between A.M., G.M. and H.M., and sums written in sigma notation.

Key concepts: Sequences and Series

Sequence and progression
A sequence {tₙ} is a set of numbers arranged in order; finite or infinite. A sequence following a specific rule is a progression. In an A.P., tₙ₊₁ − tₙ is constant.
Geometric progression
A sequence where tₙ₊₁ / tₙ = r is constant for all n. r is the common ratio. The nth term is tₙ = a rⁿ⁻¹ (a ≠ 0, r ≠ 0).
Writing terms in G.P.
Three terms: a/r, a, ar. Four terms: a/r³, a/r, ar, ar³ (ratio r²). Five terms: a/r², a/r, a, ar, ar².
Sum of n terms of a G.P.
Sₙ = a(1 − rⁿ)/(1 − r) when r < 1 and Sₙ = a(rⁿ − 1)/(r − 1) when r > 1. If r = 1, Sₙ = na.
Sum to infinity
If |r| < 1, rⁿ → 0, so S∞ = a/(1 − r). Recurring decimals such as 0.333... can be written as fractions this way.
A.M., G.M. and H.M.
For two positive numbers x and y: A = (x + y)/2, G = √(xy), H = 2xy/(x + y). G² = AH and A ≥ G ≥ H, with equality only when x = y.
Sigma notation
Σ is the summation sign. Σr = n(n + 1)/2, Σr² = n(n + 1)(2n + 1)/6, Σr³ = [n(n + 1)/2]², and Σk = nk for a constant k.

Quick revision: important points

Practice MCQs with answers

  1. The common ratio of the G.P. 3, 6, 12, 24, ... is:

    1. 3
    2. 2
    3. 1/2
    4. 6
    Show answer

    Answer: B. 2
    r = 6/3 = 2.

  2. The 6th term of the G.P. 2, 6, 18, ... is:

    1. 162
    2. 486
    3. 1458
    4. 243
    Show answer

    Answer: B. 486
    t₆ = 2 × 3⁵ = 486.

  3. The sum of the infinite G.P. 1 + 1/2 + 1/4 + ... is:

    1. 1
    2. 3/2
    3. 2
    4. Infinite
    Show answer

    Answer: C. 2
    S∞ = 1/(1 − 1/2) = 2.

  4. The G.M. of 4 and 9 is:

    1. 6.5
    2. 6
    3. 36
    4. 13
    Show answer

    Answer: B. 6
    G = √(4 × 9) = 6.

  5. The H.M. of 4 and 12 is:

    1. 8
    2. 6
    3. 7
    4. 16
    Show answer

    Answer: B. 6
    H = 2(4)(12)/(4 + 12) = 96/16 = 6.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters