Sequence and series-1

A progression is a sequence whose terms follow a certain pattern i.e., the terms are arranged under a definite rule.

Example: 1, 3, 5, 7, 9, …….. is a progression .

Types of Progression:

1. Arithmetic progression

2. Geometric progression

3. Arithmetico–geometric progression

4. Harmonic progression

5. Miscellaneous progressions

Arithmetic Progression (A.P.)

Definition:

A sequence of numbers <t_n> is said to be in arithmetic progression (A.P.) when the difference 

t_n–t_n–1  is a constant for all n ∈ N.

This constant is called the common difference of the A.P. (denoted by the letter d)

If ‘a’ is the first term and ‘d’ the common difference, then an A.P. can be represented as 

a. a + d, a + 2d, a + 3d...................

Example: 2, 7, 12, 17, 22, …… is an A.P. whose first term is 2 and common difference 5.

Algorithm to determine whether a sequence is an A.P. or not.

Step I:    Obtain t_n   (the n^th term of the sequence).

Step II:  Replace n by n – 1 in t_n to get t_n–1.

Step III: Calculate t_n – t_n–1

If   t_n–t_n–1   is independent from n, the given sequence is an A.P. otherwise it is not an A.P.

t_n = a + (n–1)d represents the n^th term of an A.P. with common difference d.

General term of an A.P.

1. Let ‘a’ be the first term and ‘d’ be the common difference of an A.P. Then its n^th term is

t_n = a + (n – 1)d 

2. p^th term of an A.P. from the end : Let ‘a’ be the first term and ‘d’ be the common difference of an A.P. having n terms. Then p^th term from the end is  (n – p + 1)yh term from the beginning

i.e.,   t_(n–p+1) = a + (n – p)d

If last term of an A.P. is l then pth term from end l – (p – 1)d

Selection of terms in an A.P.

When the sum is given, the following way is adopted in selecting certain number of terms:

When the sum is not given, then the following way is adopted in selection of terms.

Sum of n terms of an A.P.

The sum of n terms of the series

Arithmetic Mean

If a, A, b are in A.P., then A is called A.M. between a and b.

1. If  a, A1, A2, A3.......An, b are in A.P., then   A1, A2, A3.......An  are called n A.M.’s between a and b.

2. Insertion of arithmetic means

1. Single A.M. between a and b: If a and b are two real numbers then single A.M. between a and b  

=  (a+b)/2

2. n A.M.’s between a and b: If A1, A2, A3.......An  are n A.M.’s between a and b, then