Mathematics Sequences and Series

Sequences

• A sequence is an ordered arrangement of numbers following a specific rule or pattern.
• Examples of sequences include: population growth, bank deposits, or depreciated values of a commodity.
• Terms in a sequence are denoted by a₁, a₂, a₃, ... , a_n, where the subscript indicates the position of the term.
• The nth term is called the general term and is represented by a_n.
• A sequence with a finite number of terms is called a finite sequence, while a sequence that continues indefinitely is called an infinite sequence.

Series

• A series is the sum of the terms in a sequence: a₁ + a₂ + a₃ + ... + a_n + ...
• A series is called finite or infinite depending on whether the corresponding sequence is finite or infinite.

Arithmetic Progression (A.P.)

• In an arithmetic progression (A.P.), the difference between any two consecutive terms is constant.
• This constant difference is called the common difference, denoted by 'd'.
• The general form of an A.P. is: a, a + d, a + 2d, a + 3d, ... , a + (n-1)d

Geometric Progression (G.P.)

• In a geometric progression (G.P.), the ratio of any two consecutive terms is constant.
• This constant ratio is called the common ratio, denoted by 'r'.
• The general form of a G.P. is: a, ar, ar², ar³, ... , ar^n-1

Fibonacci Sequence

• This sequence is defined by the following recurrence relation:
  • a₁ = a₂ = 1
  • a_n = a_n-2 + a_n-1, for n > 2
• The first few terms of the Fibonacci sequence are: 1, 1, 2, 3, 5, 8, ...

Arithmetic Mean (A.M.)

• The arithmetic mean of two numbers 'a' and 'b' is given by: (a + b)/2

Geometric Mean (G.M.)

• The geometric mean of two positive numbers 'a' and 'b' is given by: √(ab)
• The geometric mean of a set of 'n' numbers is the nth root of the product of the numbers.

Important Notes

• Sequences and series are fundamental concepts in mathematics and have broad applications in various fields, including finance, physics, and computer science.
• Understanding the different types of sequences and series and their properties is essential for solving problems related to growth, decay, and patterns.

Explore the concepts of sequences and series including definitions and examples of finite and infinite sequences. Learn about arithmetic progression and how to identify and calculate terms within these mathematical structures.