Series and sequence | free study material for IIT JEE , NDA

Sequences and Series

What is AP ?
When the sequences or series number increase or decreases by the fixed quantity , then the sequence or series is known Arithmetic progression.
General Expression of A.P. :
a , a+d , a + 2d , a+3d ......... where a is the first term and d is common difference
nth term (a_n) : It represent the nth term of an A.P. , and it can be write down as
a_n = a + (n-1)d
NOTE : The difference between consecutive terms is always d
Sum of first n terms of an A.P. S_n = a + (a+d) + (a+2d) + ....... + (a + (n-1)d)


Arithmetic Mean (AM) :
When three numbers are in A.P. , then middle one is called arithmetic mean of other two. If a and b are two number and AM is the arithmetic mean of a and b , it means a , AM , b are in A.P.

How to insert n arithmetic means between two numbers
AM₁ , AM₂ , AM₃ .....AM_n are n arithmetic means between two numbers a and b , So the sequence is a , AM₁ , AM₂ , AM₃ .....AM_n , b are in A.P.
First term is a , number of terms are n + 2 , and last term = b
Let d be the common difference of given A.P. , then
a_n+2 = b = a + [(n+2) -1]d
b = a + (n+1)d


Important point :
1. If a , b , c are in A.P. :
    a) ak , bk , ck are also in A.P.
    b) a/k , b/k , c/k are also in A.P.
    c) a ± k , b ±k , c ± k are also in A.P.
2. Three terms of an A.P. can be taken as : a - d , a, a + d
3. Four terms of an A.P. can be taken as : a - 3d , a - d , a + d , a + 3d

What is G.P. ?
If the series or sequence of numbers a₁ , a₂ , a₃....... decrease or increase by some constant factor , they are called "Geometric Progression (G.P.)"
The general expression of Geometric progression is
a ,ar , ar² , ar³ ..... where
a is the first term
r is the common ratio
nth term (a_n) : It represent the nth term of an G.P. , and it can be write down as
a_n = ar^n-1
Sum of first n terms of an G.P. :
S_n = a + ar + ar² + ....... + ar^n-1

Geometric Mean (GM) : When three numbers are in G.P. , then middle one is called geometric mean of other two. If a and b are two number and GM is the geometric mean of a and b , it means a , GM , b are in G.P.

How to insert n geometric means between two numbers
GM₁ , GM₂ , GM₃ .....GM_n are n geometric means between two numbers a and b , So the sequence is a , GM₁ , GM₂ , GM₃ .....GM_n , b are in G.P.
First term is a , number of terms are n + 2 , and last term = b
b = (n+2)^th term
b = arⁿ⁺¹


Infinte geometric series :
If common ratio r lies between -1 and 1 , the sum of infinity terms of such G.P. :

Important point :
1. If a , b , c are in G.P. :
    a) ak , bk , ck are also in G.P.
    b) a/k , b/k , c/k are also in G.P.
2. Three terms of a G.P. can be taken as : ar , a, a/r
3. Four terms of an G.P. can be taken as : ar³ , ar , a/r , a/r³

What is H.P. ?
A sequence a₁ , a₂ , a₃ ...... is said to be in Harmonic progression if

it means ,
Harmonic Mean :
When three numbers are in H.P. , then middle one is called harmonic mean of other two. If a and b are two number and HM is the harmonic mean of a and b , it means a , HM , b are in H.P.


How to insert n harmonic means between two numbers
HM₁ , HM₂ , HM₃ , ........ ,HM_n are called n harmonic means between two number a and b
So the sequence is : a, HM₁ , HM₂ , HM₃ , ........ ,HM_n , b are in H.P.

For this AP , first term = 1/a , number of terms = n + 2 , and last term = 1/b


Relation Between AM , GM and HM



GM² = AM * HM
Note : AM > GM > HM

1. Sum of first n natural numbers

2. Sum of squares of first n natural numbers

3. Sum of cubes of first n natural naumbers :

4. Arithmetic geometric Series Sum :
It is the combination of AP and GP , the general expression of AGP :
a ,(a + d)r , (a + 2d)r² ,.....

Lecture 1 For Progression :

Lecture 2 For Progression :

Lecture 3 For Progression :

Lecture 4 For Progression :

Lecture 5 For Progression :

Lecture 6 For Progression :

Practice set 1 Practice set 2 Practice set 3