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 n^th term of an A.P. with first term = a and common difference = d is given bynt_n = a + (n − 1)d

Sum of n terms of an A.P.
→ Let S_n denote the sum of n terms of an A.P.,
then S_n = a + (a + d) + (a + 2d) + ........ a + (n - 1)d  → (1)
Also S_n = a + (n - 1)d + a + (n - 2)d + ....... (a + d) + a  → (2)
Adding (1) & (2)
2 S_n = n[2a + (n - 1)d]
 S_n =   [2a + (n - 1)d] =   [a + a + (n - 1)d]
=  (a + t_n) =  (a + l)
=

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, ......, b_n is an A.P. with common difference = d₂, then
(i) a₁ + b₁, a₂ + b₂, a₃ + b₃, ........, a_n + b_n is
     an A.P. with common difference = d₁ + d₂.
(ii) a₁ - b₁, a₂ - b₂, a₃ - b₃, ......., a_n - b_n
      is an A.P. with common difference = d₁ - d₂.
→ a₁ + a_n = a₂ + a_{n − 1} = a₃ + a_{n - 2} = ........ = ar + a_{n − 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, ......., a_n is 
Insertion of "n" Arithemetic Means between two positive numbers
→ If "n" numbers are introduced between a and b such that a, A₁, A₂, ......., A_n, b form an A.P., then A₁, A₂, ......., A_n are known as n A.M.'s between a and b.
→ T_n + 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 t_n = a r^{n − 1}
Sum of 'n' terms of a G.P.
→ Let S_n denote the sum of 'n' terms of a G.P., then
S_n = a + ar + ar² + ...... + ar^{n − 2} + ar^{n − 1}  → (1)
rS_n = ar + ar² + ....... + ar^{n − 1} + arⁿ→  (2)
subtracting (2) from (1)
S_n (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 = 

→ If a₁, a₂, a₃, ........, an are in G.P. with common ratio 'r₁' and b₁, b₂, b₃, .........., b_n are in G.P. with comman ratio 'r₂', then
i) a₁b₁, a₂b₂, a₃b₃, ........., a_nb_n
    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 a_n 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, ....., a_n is (a₁a₂a₃ ..... a_n)^1/n
Insertion of n Geometric means between two positive numbers
Let a, G₁, G₂, G₃, ........, G_n, b be in G.P., then G₁, G₂, G₃, ......., G_n are known as the Geometric means between a and b.

Harmonic Progression
→ A sequence {a₁, a₂, a₃, ........., a_n} is said to be a H.P. if   are in A.P. (a₁, a₂, a₃, ....... a_n 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, ......., a_n is

      
Insertion of 'n' Harmonic Means between a and b
 If a, H₁, H₂, H₃, ......., H_n, b are in H.P., then H₁, H₂, H₃, ........, H_n are known as the Harmonic Means between a and b.
 If a, H₁, H₂, ......., H_n, 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](br^{n − 1})
 ab + (a + d)(br) + (a + 2d)(br²) + ....... is known as an arithmetico geometric series.

Sum of 'n' terms of an A.G.P.
 S_n = ab + (a + d)(br) + (a + 2d)(br²) + ..... + [a + (n − 2)d]br^{n − 2} +
             [a + (n - 1)d]br^{n − 1}→  (1)
    rS_n = abr + (a + d)br² + (a + 2d)br³ +.....
              [a + (n - 2)d]br^{n− 1} + [a + (n - 1)d]brⁿ  → (2)
Subtracting (2) from (1)
(1 - r)S_n = ab + (dbr + dbr² + ....... + dbr^{n − 1}) - [a + (n − 1)d]brⁿ
 (1 - r)S_n = ab + dbr (1 + r + r² + ....... + r^{n − 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 cubes of first 'n' natural numbers

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

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

Subtracting
0 = x₁ + [(x₂ - x₁) + (x₃ − x₂) + ..... + (x_n − x_{n − 1})] - x_n
 x_n = x₁ + t₁ + t₂ + t₃ + ....... + t_{n − 1}
x_n = x₁ + S_{n − 1}
Where S_{n − 1} is the sum of the series t₁ + t₂ + t₃ + ....... + t_{n − 1}
which is A.P. or G.P.
S = Σx_n