Sequence and series

 Sequence:

A  sequence set that follows a definite pattern or or have a common relationship between its elements.

Element of a sequence is called terms which are separated by commas( , ).

Sequence is also called progression.

Example:

1,2,3,4,5,6-------

1,2,3,4,5,6

Real sequence:

A sequence in which all terms are real.

i is not involved.

May be finite /infinite

Complex/imaginary sequence:

A sequence in which term involved i

2, 2+I, 2+2i,--------

i, 2i ,3i ,4i, ---------

may be finite/infinite.

Series:

A series is a sum of a sequence of terms

That is, a series is a list of numbers with addition

operations between them.

Example:

2+4+6+8+,---------------

1+2+3+4+5

Types of Sequence:

Some of the most common examples of sequences are:

Arithmetic Sequences

Geometric Sequences

Harmonic Sequences

Fibonacci Numbers

Arithmetic Sequence

A mathematical sequence in which the difference between two  consecutive terms is always a constant and has a form:

                   a,a+d,a+2d……..a+(n-1)d

Graphically behaves as linear function.

Notation In AP

In Arithmetic sequence, we come across two main terms which are denoted as :

Common difference(d)

Nth term(an)

All three terms represents the properties of 

arithmetic sequence or arithmetic progression.

Formula For Nth Term Of AP

The formula for finding the nth term is:

                    an = a + (n-1) * d

        Where,

                              a=first term

                             d= common difference 

                             n=number of terms

                              an=nth term

Example

Find the nth term of AP:

                1,2,3,4,5……….,an     if number of terms are 15.

Solution :

                     by the formula we know 

                        an=a+(n-1)d------------(1)

                                n=15

                                d=a2-a1

                                      =2-1=1

                        putting in equation (1)

                                =   1+(15-1)1

                                     =   15

                                                                     

                                        

Arithmetic Means (A.Ms):

If a, A, b are three consecutive terms in an Arithmetic Progression,

Then A is called the Arithmetic Mean (A.M) of a and b.

i.e. if a, A, b are in A.P. then

A – a = b – A

A + A = a + b

2A = a + b

A =a + b/2

Arithmetic Series:

The sum of the terms of an Arithmetic sequence is called as

Arithmetic series. For example:

7, 17, 27, 37, 47, - - - - - - - - - is an A.P.

7 + 17 + 27 + 37 + 47 + - - - - - - - - is Arithmetic series.

Geometric Sequence

A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio r.”

    It is some time called geometric progression            or GP

Common Ratio

The common ratio is obtained by dividing any term by the preceding term i.e.,

                                         

r=a2/a1=a3/a2=………an/an-1

r= common ratio 

a1=first term

a2=second term

a3=third term 

an-1=the term before nth term

an=nth term

Formula For Nth Term

To find the nth term of GP we use the formula 

                          an=a1r^n-1

    Where

            r=common ratio

           a1=first term

           an-1=the term before the nth term

           n=number of terms

Geometric mean

When three quantities are in G.P., the middle one is called the

Geometric Mean (G.M.) between the other two.

Geometric Series

A geometric series is the sum of the terms of a geometric

sequence.

If a, ar, ar2, …… + arn-1 is a geometric sequence.

Then a + ar + ar2+ …… + arn-1 is a geometric series.

Harmonic progression:

A series of numbers is said to be in harmonic

sequence if the reciprocals of all the elements

of the sequence form an arithmetic sequence.

Example:

           1,1/4, 1/7, 1/10, -----   &  1+1/4+1/7+1/10+------

A sequence in the form:

          1/a ,1/a+d, 1/a+2d, 1/a+3d,-----

Sum of seies:

             Sn =1/ [a + (n -1) d].

Harmonic mean:

Find harmonic mean of 2, 4, 6 and 100.

The arithmetic mean is,

2+4+6+100/4 = 28

The harmonic mean is,

= 4/(1/2 + 1/4 + 1/6 + 1/100)

= 4.32 (to 2 places)

Harmonic series:

A harmonic series is the sum of the terms of  harmonic sequence.

If 1,1/4,1/7,1/10,1/13……… is a harmonic sequence

Then,

1+1/4+1/7+1/10+………….. is a harmonic series.

Fibonacci sequence

       

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

The next number is found by adding up the two numbers before it:

the 2 is found by adding the two numbers before it (1+1),

the 3 is found by adding the two numbers before it (1+2),

the 5 is (2+3),

and so on!

If 4,7,10,13,16,19,22……is a sequence, Find:

Common difference

nth term

Solution:

21st termGiven sequence is, 4,7,10,13,16,19,22……

a) The common difference = 7 –  4 = 3

b) The nth term of the arithmetic sequence is denoted by the term Tn and is given by Tn = a + (n-1)d, where “a” is the first term and d, is the common difference.

Tn = 4 + (n – 1)3 = 4 + 3n – 3 = 3n + 1

c) 21st term as:  T21 = 4 + (21-1)3 = 4+60 = 64.

Question 2:Question 1:

Consider the sequence 1,4,16,64,256,1024….. Find the common ratio and 9th term.

Solution: The common ratio (r)  = 4/1 = 4

The preceding term is multiplied by 4 to obtain the next term.

The nth term of the geometric sequence is denoted by the term Tn and is given by Tn = ar(n-1) where a is the first term and r is the common ratio.

Here a = 1, r = 4 and n = 9

So, 9th term is can be calculated as T9 = 1* (4)(9-1)= 48 = 65536

Question 3

Write the first five terms of a sequence described by the general term a n = 3 n + 2.

A1=3(1)+2=5

A2=3(2)+2=8

A3=3(3)+2=11

A4=3(4)+2=14

a5=3(5)+2=17

Therefore, the first five terms are 5, 8, 11, 14, and 17.