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Sequences and Series – Class 11 Mathematics notes, important questions and MCQs

NCERT Class 11 Mathematics, chapter 8. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Sequences are ordered lists of numbers; series are their sums. Arithmetic and geometric progressions have simple formulas, and AM ≥ GM links them.

AP and GP

AP: aₙ = a + (n − 1)d; Sₙ = n/2[2a + (n − 1)d].
GP: aₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1; infinite sum S = a/(1 − r) for |r| < 1.
Arithmetic mean A = (a + b)/2; geometric mean G = √(ab); A ≥ G.

Special sums

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

Key terms

AP: sequence with a constant difference
GP: sequence with a constant ratio
Arithmetic mean: the number A between a and b with a, A, b in AP; A = (a + b)/2

Formulas

Sₙ = n/2[2a + (n − 1)d]
S∞ = a/(1 − r)
Σn² = n(n + 1)(2n + 1)/6

Important questions with answers

Find the 20th term of the AP 7, 4, 1, …2 marks

a = 7, d = −3. a₂₀ = 7 + 19(−3) = −50.

The 3rd and 6th terms of a GP are 24 and 192. Find the first term and the common ratio.3 marks

ar² = 24 and ar⁵ = 192, so r³ = 8, r = 2 and a = 24/4 = 6.

Find the sum to n terms of the series 1 × 2 + 2 × 3 + 3 × 4 + …, and its value for n = 10.5 marks

nth term = n(n + 1) = n² + n. Sₙ = Σn² + Σn = n(n + 1)(2n + 1)/6 + n(n + 1)/2 = n(n + 1)(2n + 4)/6 = n(n + 1)(n + 2)/3. For n = 10: 10 × 11 × 12/3 = 440.

Quick revision

AM ≥ GM.
Infinite GP sum needs |r| < 1.
Σn³ = (Σn)².

Exam tips

Three numbers in AP: take a − d, a, a + d; in GP: a/r, a, ar.
For "sum to n terms" of special series like 1·2 + 2·3 + …, write the nth term and use Σn, Σn², Σn³.

Practice MCQs

10th term of the AP 2, 5, 8, … is
1. 27
2. 32
3. 30
4. 29
Show answer
Answer: (d) 29. 2 + 9 × 3.
Common ratio of the GP 3, 6, 12, … is
1. 3
2. 2
3. 6
4. ½
Show answer
Answer: (b) 2. 6/3.
Sum of the first 20 natural numbers is
1. 200
2. 400
3. 210
4. 190
Show answer
Answer: (c) 210. 20 × 21/2.
Sum to infinity of 1 + ½ + ¼ + … is
1. 1
2. 1.5
3. 2
4. ∞
Show answer
Answer: (c) 2. 1/(1 − ½).
Sum of the first 10 terms of the GP 1, 2, 4, … is
1. 2047
2. 1024
3. 511
4. 1023
Show answer
Answer: (d) 1023. 2¹⁰ − 1.
GM of 4 and 16 is
1. 10
2. 8
3. 64
4. 12
Show answer
Answer: (b) 8. √64.
Sum of the squares of the first 10 natural numbers is
1. 330
2. 385
3. 505
4. 285
Show answer
Answer: (b) 385. 10 × 11 × 21/6.
The sum 1 + 3 + 6 + 10 + … to 10 terms is
1. 165
2. 220
3. 286
4. 110
Show answer
Answer: (b) 220. Σ n(n + 1)/2 for n = 1 to 10.

All chapters of Class 11 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.