NCERT Mathematics Class 11 - Chapter 8: Sequences and Series - Notes

Learning Objectives

Understand arithmetic progressions (AP) and their properties
Learn geometric progressions (GP) and their properties
Study the relationship between AM, GM, and HM
Learn the sum of special series

Key Concepts

Arithmetic Progression (AP)

Sequence where the common difference d = a_n - a_n-1 is constant. General term: a_n = a + (n-1)d. Sum of n terms: S_n = n/2 [2a + (n-1)d] = n/2 [a + l] where l = last term. nth term from sum: a_n = S_n - S_n-1.

Properties: If a, b, c are in AP then 2b = a + c (b is the arithmetic mean). AM of a and b = (a + b)/2. Three terms in AP: a - d, a, a + d. Four terms: a - 3d, a - d, a + d, a + 3d. Sum of n terms of AP is a quadratic function of n.

Geometric Progression (GP)

Sequence where the common ratio r = a_n/a_n-1 is constant. General term: a_n = ar^n-1. Sum of n terms: S_n = a(rⁿ - 1)/(r - 1) for r ≠ 1; S_n = na for r = 1. Sum to infinity (|r| < 1): S_∞ = a/(1 - r).

Properties: If a, b, c are in GP then b² = ac (b is the geometric mean). GM of a and b = √(ab). Three terms in GP: a/r, a, ar. Product of n terms of GP with first term a and last term l: (al)^n/2.

Harmonic Progression (HP)

Sequence where reciprocals form an AP. Example: 1, 1/2, 1/3, 1/4, ... No direct formula for sum. Harmonic Mean: HM of a and b = 2ab/(a + b).

Relationship: AM, GM, HM

For two positive real numbers a and b: AM ≥ GM ≥ HM. AM = (a+b)/2, GM = √(ab), HM = 2ab/(a+b). Equality holds when a = b. Also, AM × HM = GM².

Sum of Special Series

Σk (k=1 to n) = n(n+1)/2. Σk² = n(n+1)(2n+1)/6. Σk³ = [n(n+1)/2]² = (Σk)². Sum of first n natural numbers, squares, and cubes are frequently tested in JEE.

Summary

AP has constant common difference; GP has constant common ratio. Key formulas exist for the nth term and sum of terms. AM ≥ GM ≥ HM for positive numbers. Special series sums (natural numbers, squares, cubes) are important for JEE.

Important Terms

AP: Sequence with constant common difference d
GP: Sequence with constant common ratio r
Arithmetic Mean: AM = (a + b)/2
Geometric Mean: GM = √(ab)
Harmonic Mean: HM = 2ab/(a + b)
Infinite GP sum: S∞ = a/(1-r) when |r| < 1

Quick Revision

AP: a_n = a + (n-1)d; S_n = n/2[2a + (n-1)d]
GP: a_n = ar^n-1; S_n = a(rⁿ-1)/(r-1)
Infinite GP: S∞ = a/(1-r), |r| < 1
AM ≥ GM ≥ HM; AM × HM = GM²
Σk = n(n+1)/2; Σk² = n(n+1)(2n+1)/6; Σk³ = [n(n+1)/2]²
In AP: 2b = a + c; In GP: b² = ac
Three AP terms: a-d, a, a+d; Three GP terms: a/r, a, ar