HSSLIVE Plus One Maths Chapter 9: Sequences and Series Notes

This chapter examines ordered collections of numbers (sequences) and the sums of their terms (series). Students will explore arithmetic and geometric sequences, special series, convergence properties, and methods for finding sums of infinite series. Through this study, students develop pattern recognition skills and learn mathematical tools that model growth processes, financial calculations, and physical phenomena. These concepts form the foundation for calculus and are essential for understanding functions as infinite series.

Chapter 9: Sequences and Series

A sequence is an ordered list of numbers following a pattern, while a series is the sum of the terms of a sequence.

Arithmetic Progression (AP):

A sequence where each term differs from the preceding term by a constant value (common difference).

• General Form: a, a+d, a+2d, a+3d, …
• nth Term: aₙ = a + (n-1)d
• Sum of n Terms: Sₙ = n/2[2a + (n-1)d] = n/2(a + l) where l is the last term

Geometric Progression (GP):

A sequence where each term is a constant multiple (common ratio) of the preceding term.

• General Form: a, ar, ar², ar³, …
• nth Term: aₙ = ar^(n-1)
• Sum of n Terms: Sₙ = a(1-r^n)/(1-r) for r≠1
• Sum of Infinite GP: S∞ = a/(1-r) for |r| < 1

Harmonic Progression (HP):

A sequence where the reciprocals of the terms form an arithmetic progression.

• General Form: If aₙ forms an HP, then 1/aₙ forms an AP.

Arithmetic Mean (AM):

AM between a and b is (a+b)/2

Geometric Mean (GM):

GM between a and b is √(ab)

Harmonic Mean (HM):

HM between a and b is 2ab/(a+b)

Special Series:

1. Sum of n natural numbers: 1 + 2 + 3 + … + n = n(n+1)/2
2. Sum of squares: 1² + 2² + 3² + … + n² = n(n+1)(2n+1)/6
3. Sum of cubes: 1³ + 2³ + 3³ + … + n³ = [n(n+1)/2]²