In this session explained about basic concepts of sequence and series with their problems and solutions

## Basic Concepts of Sequence and Series Problems and Solutions

Contents

### Sequence

A set of numbers arranged in a definite order according to some definite rule is called* sequence*.

i.e A *sequence* is a set of numbers written in a particular order.

Now take a sequence

m_{1}, m_{2}, m_{3}, m_{4, } . . . . . . . . , m_{n}

Here first term in a sequence is m_{1}, the second term m_{2}, and so on. With this same notation, n^{th} term in the sequence is m_{n}.

**Examples of sequence
**

1. Take the numbers 1, 5, 9, 13, 17, . . . . . .

In the above numbers seem to have a rule. It is started with the a number 1, and add a number 4 to previous number to obtained successive number

2. Take the numbers 1, 5, 25, 125, 625 . . . . . .

In the above numbers seem to have a rule. It is started with the a number 1, and multiplying with a number 5 to previous number to obtained successive number

3. Take the numbers 1, 8, 27, 64, 125, 216 . . . .

This is the sequence of cube of numbers.

4. Take the numbers 2 , −2, 2, −2, 2, −2, . . .

In the above sequence of numbers alternating between 2 and −2.

In above all examples, the dots written at the end indicate that we must consider the sequence as an** infinite sequence.**

5. The numbers 1, 9, 17, 25, 33

In the above example having form a finite sequence containing just five numbers.

6. Take numbers 1, 3, 5, 7, 9, 11, . . . . . . . n

In the above sequence, last number mentioned as ‘n’. So it is finite sequence

### Series

The sum of the terms of a sequence is called a series

i.e A series is obtained from a sequence by adding all the terms together.

Take a sequence m_{1}, m_{2}, m_{3}, m_{4} . . . . . . . . , m_{n}

The series we obtain from this is m_{1 }+ m_{2}+ m_{3 }+ m_{4} + . . . . . . . . + m_{n}

**Finite series and Infinite series**

The series 1 + 9 + 17 + 25 + 33 is called* finite series*

The series 1 + 9 + 17 + 25 + 33+ . . . . . . . . . . . . is called *infinite series*

#### Sequence and series examples

1. A sequence is given by the formula m_{n} = 2n + 3, for n = 1, 2, 3, . . .. now find first four terms of this sequence.

**Solution**: n^{th} term of the sequence is m_{n} = 2n + 3

First term of the sequence is = 2 ( 1)+3 = 5 ( substitute n = 1)

Second term of the sequence is = 2 ( 2 )+3 = 7 ( substitute n = 2)

Third term of the sequence is = 2 ( 3 )+3 = 9 ( substitute n = 3)

Fourth term of the sequence is = 2 ( 4)+3 = 11 ( substitute n = 4)

2. A sequence is given by m_{n} = 2/(n+2), for n = 0, 1, 2, 3, . . .then find 8th term?

**Solution**: n^{th} term of the sequence is m_{n} = 2/(n+2)

Now 8th term of the sequence is (1.e n= 7) = 2/9

3. Find first three terms of the following sequence, beginning with n = 0,1 & 3. m_{n} = 2/(n+2)!

**Solution:** n^{th} term of the sequence is m_{n} = 2/(n+2)!

Now 1st term of the sequence is (1.e n= 0) = 2/2! = 2/2 = 1

2nd term of the sequence is (1.e n= 1) = 2/3! = 2/6 = 1/3

3rd term of the sequence is (1.e n= 2) = 2/4! = 2/24 = 1/12

So the sequence is 1 , 1/3 , 1/12

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