Sequences

Definition (Sequence):

A set of numbers arranged in a definite order according to some definite rule (or rules) is called a sequence. Each number of the set is called a term of the sequence. A sequence is called finite or infinite according as the number of terms in it is finite or infinite. The different terms of a sequence are usually denoted by or by .The subscript should be a natural number denoting the position of the term in the sequence. The number occurring at the nth place of a sequence i.e. is called the general term of the sequence.

A finite sequence is described by or by and an infinite sequence is described by or . If all the terms are real, we have a real sequence; if all terms are complex numbers, we have a complex sequence etc.

For example, let us consider the following sequence:

1, 3, 5, 7,…, 21.

In the above sequence each term is obtained by adding 2 to the previous term. Also, this is a finite sequence. To define a sequence, we need not always have an explicit formula for the nth term. For the above sequence we may write

If the terms of a sequence can be described by an explicit formula, then the sequence is called a progression.

Note: The sequence 1, 1, 2, 3, 5, 8, 13,…., is also a progression. It is called Fibonacci sequence.

Definition (Series):

If the terms of a sequence are connected by plus signs we get a series. Thus, if is a given sequence then the expression is called the series associated with the given sequence. The series is finite or infinite according as the given sequence is finite or infinite.

If denotes the general term of a sequence, then is a series of n terms. In a series , the sum of first n terms is denoted by . Thus,

If denotes the sum of terms of a sequence, then

Thus, .

Example 1: Find the next term of the sequence

(i) 2, 4, 6, 8

(ii) 2, 8, 32, 128

(iii) -1, -3, -5, -7

(iv) 1, 8, 27, 64

Solution:

(i) We see that each term is obtained by adding 2 to the previous term. Hence next term= 8+ 2= 10.

(ii) We see that each term is obtained by multiplying the previous term by 4. Hence next term

(iii) We see that each term is obtained by subtracting 2 from the previous term. Hence next term= -7-2 = -9

(iv) We see that terms are cubes of natural numbers – . Hence next term =

Example 2: Find the 18^th term of the sequence defined by .

Solution:

Here , putting , we get,

.

Example 3: Find the first five terms of the sequence given by $latexz a_1= 2, a_2=3+a_1 $ and for .

Solution:

Here

Given for , putting =3, 4, 5 we get

Hence the first five terms of the given sequence are 2, 5, 15, 35, 75.

Example 4: If for a sequence , find its first four terms.

Solution:

Given,

Putting we get

and

Hence the first four terms of the sequence are 4, 12, 36, 108.

Example 5: Write in expanded form.

Solution:

Putting k = 1, 2,3, 4, … , n in , we get 2, 5, 10, 17, … , n^2+1

Hence .

Exercise:

1) Write the first five terms of the sequences defined by

(i)

(ii)

2) Find the 18^th and 25^th terms of the sequence defined by

, if is even natural number

, if is odd natural number.

3) A sequence is defined by . Show that first two terms of the sequence are zero and the rest of the terms are positive.

4) The Fibonacci sequence is defined by for . Find for .

5) First term of a sequence is 1 and the term is obtained by adding to the term for all natural numbers . Find the sixth term of the sequence.

6) Find the first five terms of the sequence defined by

(i) for

(ii) where is the sum of terms.

(iii) for