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SEQUENCES AND SERIES

Sequence

A sequence is a function whose domain is the set of natural numbers and whose codomain is the set of real or complex numbers.
A sequence whose codomain is the set of real numbers is known as a real sequence.
A sequence whose codomain is the set of complex numbers is known as a complex sequence.

 

Arithmetic Progression
A sequence in which the terms increase or decrease by a fixed number is known as an Arthmetic Progression. This fixed number is known as the common difference of the A.P.
An A.P. with first term = a and common difference = d can be written as a, a + d, a+ 2d, ........
The common difference of an A.P. may be positive, negative or zero.
If d > 0 increasing A.P.
    If d < 0 decreasing A.P.
    If d = 0 constant A.P.
The nth term of an A.P. with first term = a and common difference = d is given byntn = a + (n − 1)d

Sum of n terms of an A.P.
Let Sn denote the sum of n terms of an A.P.,
then Sn = a + (a + d) + (a + 2d) + ........ a + (n - 1)d   (1)
Also Sn = a + (n - 1)d + a + (n - 2)d + ....... (a + d) + a   (2)
Adding (1) & (2)
2 Sn = n[2a + (n - 1)d]
 Sn =   [2a + (n - 1)d] =   [a + a + (n - 1)d]
=  (a + tn) =  (a + l)
=  (first term + last term)


Properties of an A.P.
If a1, a2, a3, ........, an are in A.P. with common difference = d, then
(i) a1 + k, a2 + k, a3 + k, ......., an + k is an A.P. with common difference = d
(ii) a1 - k, a2 - k, a3 - k, ......., an - k is an A.P. with common difference = d.
(iii) ka1, ka2, ka3, ......., kan is an A.P. with common difference = kd.



If a1, a2, a3, ......., an is an A.P. with common difference = d1 and b1, b2, b3, ......, bn is an A.P. with common difference = d2, then
(i) a1 + b1, a2 + b2, a3 + b3, ........, an + bn is
     an A.P. with common difference = d1 + d2.
(ii) a1 - b1, a2 - b2, a3 - b3, ......., an - bn
      is an A.P. with common difference = d1 - d2.
a1 + an = a2 + an − 1 = a3 + an - 2 = ........ = ar + an − r + 1.
Three terms in A.P. can be chosen as a - d, a, a + d, four terms in A.P. can be
     chosen as a - 3d, a - d, a + d, a + 3d.


Arithmetic Mean
If a, A, b are in A.P., then A is known as the Arithmetic Mean of a and b.

  A − a = b - A
 2A = a + b   A = 

The Arithmetic Mean of "n" numbers a1, a2, a3, ......., an is 
Insertion of "n" Arithemetic Means between two positive numbers
If "n" numbers are introduced between a and b such that a, A1, A2, ......., An, b form an A.P., then A1, A2, ......., An are known as n A.M.'s between a and b.
Tn + 2 = b = a + (n + 1)d
where d = common difference.


 
Geometric Progression
A sequence in which the ratio between any term and its preceding term is a constant is known as a geometric progression.


The constant is known as the common ratio of the G.P. denoted by 'r'.
r > 1 increasing G.P.
r < 1 decreasing G.P.
r = 1 constant G.P.
A G.P. with first term = a and common ratio = r can be written as a, ar, ar2, .........
The nth term of G.P. with first term = a and common ratio = r is tn = a rn − 1
Sum of 'n' terms of a G.P.
Let Sn denote the sum of 'n' terms of a G.P., then
Sn = a + ar + ar2 + ...... + arn − 2 + arn − 1   (1)
rSn = ar + ar2 + ....... + arn − 1 + arn   (2)
subtracting (2) from (1)
Sn (1 - r) = a - arn



          = na; r = 1
Sum of Infinity of a G.P.


  
Properties of a G.P.
If a1, a2, a3, ..........., an are in G.P. with common ratio = r, then
i) ka1, ka2, ka3, ........., kan are in G.P. with common ratio =   = r

If a1, a2, a3, ........, an are in G.P. with common ratio 'r1' and b1, b2, b3, .........., bn are in G.P. with comman ratio 'r2', then
i) a1b1, a2b2, a3b3, ........., anbn
    is a G.P. with common ratio = r1r2

If a1, a2, a3, ......., an are in G.P., then log a1, log a2, log a3, ......, log an are in A.P.
Three terms in G.P. can be considered ass  , a, ar
Four terms in G.P. can be considered as 

 

Geometric Mean
If a, G, b are in G.P., then G is known as Geometric mean of a and b.

     

The Geometric mean of 'n' positive numbers a1, a2, a3, ....., an is (a1a2a3 ..... an)1/n
Insertion of n Geometric means between two positive numbers
Let a, G1, G2, G3, ........, Gn, b be in G.P., then G1, G2, G3, ......., Gn are known as the Geometric means between a and b.

Harmonic Progression
A sequence {a1, a2, a3, ........., an} is said to be a H.P. if   are in A.P. (a1, a2, a3, ....... an are non zero)

If a, a + d, a + 2d, ......., a + (n − 1)d is an A.P., then 


      
Note: We do not have any formula for sum of 'n' terms of a H.P.

 

Harmonic Mean


      

H is known as the Harmonic Mean of a and b.
The Harmonic Mean of the numbers a1, a2, a3, ......., an is

      
Insertion of 'n' Harmonic Means between a and b
 If a, H1, H2, H3, ......., Hn, b are in H.P., then H1, H2, H3, ........, Hn are known as the Harmonic Means between a and b.
 If a, H1, H2, ......., Hn, b are in H.P., then 

 
       

     
Arithmetico Geometric Progression (A.G.P.)
 If every term of a sequence is the product of two factors such that the first factors form an A.P. and the second factors form a G.P., then the sequence is known as an A.G.P.
 a.b, (a + d)(br), (a + 2d)(br2), ............. is an A.G.P.
 nth term of A.G.P. = [a + (n - 1)d](brn − 1)
 ab + (a + d)(br) + (a + 2d)(br2) + ....... is known as an arithmetico geometric series.


Sum of 'n' terms of an A.G.P.
 Sn = ab + (a + d)(br) + (a + 2d)(br2) + ..... + [a + (n − 2)d]brn − 2 +
             [a + (n - 1)d]brn − 1   (1)
    rSn = abr + (a + d)br2 + (a + 2d)br3 +.....
              [a + (n - 2)d]brn − 1 + [a + (n - 1)d]brn   (2)
Subtracting (2) from (1)
(1 - r)Sn = ab + (dbr + dbr2 + ....... + dbrn − 1) - [a + (n − 1)d]brn
 (1 - r)Sn = ab + dbr (1 + r + r2 + ....... + rn − 2) - [a + (n - 1)d]brn

           
             

Sum of infinity of an A.G.P.


                                                
Relation between AM, GM and HM
 Let A, G, H denote the Arithmetic, Geometric and Harmonic means of two positive numbers a and b, then  
 A, G, H are in G.P.
 A G H inequality holds for a = b

 

Summation of Series
 Sum of first 'n' natural numbers

 
      
 *   Sum of squares of first 'n' natural numbers

  
* Sum of cubes of first 'n' natural numbers


      
* Sum of sequences using sigma notation
Let Sn = x1 + x2 + x3 + ..... + xn - 1 + xn = Σ xn
If xn = an4 + bn3 + cn2 + dn + e, where a, b, c, d, e are constants.
Sn = Σ (an4 + bn3 + cn2 + dn + e)
= aΣn4 + bΣn3 + cΣn2 + dΣn + eΣ1

 

Method of Differences
S = x1 + x2 + x3 + ...... + xn
Let tn − 1 = xn − xn − 1
If t1, t2, t3, t4, ......., tn is an A.P. or G.P., then x1 + x2 + x3 + ....... + xn is known as a difference series.
Method of evaluate the sum
S = x1 + x2 + x3 + ..... + xn − 1 + xn
S = x1 + x2 + ....... + xn − 1 + xn

Subtracting
0 = x1 + [(x2 - x1) + (x3 − x2) + ..... + (xn − xn − 1)] - xn
 xn = x1 + t1 + t2 + t3 + ....... + tn − 1
xn = x1 + Sn − 1
Where Sn − 1 is the sum of the series t1 + t2 + t3 + ....... + tn − 1
which is A.P. or G.P.
S = Σxn

Posted Date : 19-02-2021

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