Sequence and series Introduction with examples | allmathtricks

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

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

m₁, m₂, m₃, m_4,  . . . . . . . . ,  m_n

Here first term in a sequence is m₁, the second term m₂, 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   m₁, m₂, m₃, m₄ . . . . . . . . ,  m_n

The series we obtain from this is   m₁ + m₂+ m₃ + m₄ + . . . . . . . .  + 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 & 2.  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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