Introduction
A set of numbers arranged in a definite order according to some definite rule is called a sequence.
Sequences and Series
A set of numbers arranged in a definite order according to some definite rule is called a sequence. A sequence is a function whose domain is the set N of natural numbers.
Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series.
Arithmetic Progression
Quantities are said to be in Arithmetic progression when they increase or decrease by a common difference.
Some Properties of A.P.
If a,b,c,d are in A.P., then
(ii) ka, kc, kb, kd …are also in A.P.
Arithmetic Mean (A.M.)
If a, x, b are in A.P, then x is called the arithmetic mean (A.M.) between the extremes a and b.
Geometric Progressions (G.P.)
The series of terms a, ar, ar2, ar3,.... in which each term bears a constant ratio to the preceeding term is a geometric progression. The constant ratio is called the common ratio.
Harmonic Progression (H.P.)
A sequence of numbers is said to form a harmonic progression if their reciprocals form an arithmetic progression.
Harmonic Mean (H.M.)
If three quantities are in harmonic progression, then the middle quantity is called the harmonic mean between the other two.
To insert n Harmonic Means between two given quantities
Let a and b be two given quantities. It is required to insert n harmonic means h1, h2, h3,....hn between the quantities a and b.
Arithmetic Geometric Series
A series of the form a + (a + d)r + (a + 2d)r2 + ... is called an Arithmetic-Geometric series. In the series if we put d = 0 we get GP and if we put
r = 1, we get an AP.
Sigma Notation
The Greek letter S (read as sigma) denotes the sum. When written before the nth term of series, implies, the sum of all terms obtained by giving to n the different values 1,2,3…n.
General Series
1. To find the sum of first n natural numbers.
2. To find the sum to squares of first n natural numbers.
3. To find the sum to the cubes of first n natural numbers.
4. Method of finding sum of a series whose nth term is known.
Summary
Let X be a set of numbers and f : Nn --> X be a function, then the ordered set {f(1), f(2),...., f(n)} is called a finite sequence in X.
