Sequence and Series

 

Sequence

A sequence is a function defined on set N = {1, 2, 3, 4, 5, …} of the positive integers. Since, the domain of every sequence is the set N of natural numbers, so the sequence is represented by its range.

The image of 1, 2, 3, …, n, … under the function f are generally denoted by t₁, t₂, t₃…, i.e.,    t₁ = f(1), t₂ = f(2), t₃ = f(3), …, t_n = f(n) …..

Series

If t₁, t₂, t₃, … t_n … is a sequence, with terms t₁, t₂, t₃…,t_n,... then the expression of the form

t₁ + t₂ + t₃ + …. + t_n + … is a series with terms t₁, t₂, t₃……

Progression

It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicit formula for the n^th term. Those sequences whose terms follow certain rule or follow certain pattern are called progression. The special types of progression are

i)      Arithmetic progression

ii)     Geometric progression

iii)    Harmonic Progression.

Arithmetic Progression (A.P.)

A sequence of numbers is said to be in arithmetic progression if the algebraic differences between any term and preceding term is constant through the sequence. Thus terms t₁, t₂, t₃, ... are said to be arithmetic progression if t₂ - t₁ = t₃- t₂ = t₄ - t₃ etc. The difference between any term and its preceding term is called the common difference and it is denoted by d.

For example: 1, 3, 5, 7, … are in A.P.

Then the common difference = 3 - 1 = 5 - 3 = 2.

The n^th term

Let a be the first term and d be the common difference of an A.P. Then

2^nd term t₂ = a + d = a + (2 - 1)d

3^rd term t₂ = a + 2d = a + (3 - 1) d

4^th term t₄ = a + 3d = a + (4 - 1) d

….. ….          …..

n^th tem t_n = a + (n - 1) d

Arithmetic Mean

i)      Single arithmetic mean between a and b

         Let m be an arithmetic mean between a and b.

         Therefore a, m, b are in A. P., then

        m - a = b - m

or,    2m = a + b

    

Properties of A.P.

i)      If each of the terms of an A.P. be increased or decreased by a constant quantity, the resulting quantities are in A.P. with the same common difference as before.

ii)     If each terms of A.P. be multiplied by or divided by a constant quantity, the resulting quantities are in A.P. with a common difference equal to that of given series multiplied or divided by the corresponding constant quantity.

Geometric Progression

         A sequence of numbers is said to be in geometric progression when the ratio of each term to its preceding term is always a constant quantity. The constant ratio is called the common ratio of G.P and it is denoted by r.

The n^th term

         Let first term of G.P. be a and r be the common ratio then,

         Second term t₂ = ar = ar^2-1

         Third term t₃ = ar² = ar^3-1

         Fourth term t₄ = ar³ = ar^4-1

         …  …           …

         n^th term t_n = ar^n-1