SEQUENCE AND SERIES

SEQUENCE AND SERIES
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1. DEFINITION ::

SEQUENCE :
A succession of terms a₁, a₂, a₃ ,a₄...... formed according to some rule or law.
Examples are 1,4 ,9 ,16,25
                          -1, 1,-1,1.....
                x/1! , x²/2!, x³/3!, x⁴/4!.....
It is not necessary for the terms to be unequal. A finite sequence has a finite (i.e. limited) number of terms, as in the first example above. An infinite sequence has an unlimited number of terms, i.e. there is no last term, as in the second and third examples.

SERIES :
The indicated sum of the terms of a sequence. In the case of a finite sequence a₁,a₂,a₃,........,aᵣ the corresponding series is a₁+a₂+a₃+...aᵣ = ʳ∑ᵢ aᵣ . This series has a finite or limited number of terms and is called a finite series.

2.
ARITHMETIC PROGRESSION (AP)

AP is a sequence whose terms increase or decrease by a fixed number. This fixed number is called the COMMON DIFFERENCE. If a is the first term & d the common difference, then AP can be written as a, a+d, a+2d, ......
  a+(n+d), .....

a) nᵗʰ term of this AP : :
      Tₙ= a+(n-1)d
      Where d = aₙ - aₙ₋₁
b) The sum of the first n terms ::
   Sₙ= n/2  (2a + (n -1)d) =n/2(a+l)
     where l is the last term.
c) Also nᵗʰ term Tₙ= Sₙ -  Sₙ₋₁

** NOTE **
a) Sum of n terms of an A. P. is of the form An²+Bn i.e. a quadratic expression in n, in such case the common difference is twice the coefficient of n² . i.e. 2A
b) nth term of an A. P. is of the form An+ B i.e. a linear expression in n, in such a case the coefficient of n is the common difference of the A. P. i.e. A.

3.  PROPERTIES OF A. P :

a) If each term of an A. P. is increased, decreased, multiplied or divided by the same nonzero number, then the resulting sequence is also an A. P.
b) Three numbers in A. P. can be taken as a - d, a, a+d ;
Four numbers in A. P. can be taken as a- 3d, a - d, a+ d, a + 3d.
Five numbers in A. P. a - 2d, a - d ,
  a, a + d, a + 2d .
Six terms are in A. P. a - 5d, a - 3d,
a - d, a+3d, a+5d etc.
c) The common difference can be zero, positive or negative.
d) The sum of the two terms of an AP equidistant from the beginning & end is constant and equal to the sum of first & last terms.
e) Any term of an AP (except the first) is equal to half the sum of terms which are equidistant from it.
aₙ = (1/2) (aₙ₋ₓ + aₙ₊ₓ), x < n.
For x=1, aₙ = (1/2)(aₙ₋₁ + aₙ₊₁),
For x= 2, aₙ =(1/2)(aₙ₋₂ + aₙ₊₂) and so on .
f) If a,b,c are in AP ⇒ 2b = a+c.

4. GEOMETRIC PROGRESSION(GP)

GP is a sequence of numbers whose first term is non-zero & each of the succeeding terms is equal to the proceeding terms multiplied by a constant. Thus in a GP the ratio of successive term is constant. This constant factor is called the COMMON RATIO of the series & is obtained by dividing any term by the immediately previous term. Therefore a, ar, ar², ar³, ar⁴, ... is a GP with 'a' as the first term & 'r' as common ratio.
a) nᵗʰ term Tₙ = a rⁿ⁻¹
b) Sum of the 1ˢᵗ n terms =
        Sₙ =   a(rⁿ - 1)  if r ≠ 1 and r >1
                    (  r - 1)
     And  a(rⁿ - 1)/(r - 1)  when r <1.
c) Sum of infinite GP when |r|<1
     and n -> ∞, rⁿ -> 0 S∞=a/(1-r)
                               where |r|<1

5.  PROPERTIES OF GP
a) If each term of a GP be multipled
or divided by the same non-zero, quantity, the resulting sequence is also a GP.
b) Any 3 consecutive terms of a GP can be taken as a/r, a, ar, ;
any 4 consecutive terms of a GP can be taken as a/r³, a/r, ar, ar³
& so on.
c) If a,b,c are in GP => b² = ac.
d) In an GP., the product of two terms which are at a equidistant from the first and the last term, is constant and is equal to product of first and last term.
e) If each term of a GP be raised to the same power, then resulting series is also a GP.
f) In a G. P every term (except first) is GM of its two terms which are at equidistant from it. i.e.
     tᵣ= √(Tᵣ₋ₓ Tᵣ₊ₓ), x < r.
g) In a finite G. P. the number of terms be odd then its middle term is the G. M. of the first and last term.
h) If the terms of a given G. P are choosen at right intervals, then the new sequence is also a GP.
I) If a₁,a₂ ,a₃....aₙ is a GP. of a non zero, non negative terms, then log₁log a₂,.... log aₙ is an A. P. and vice-versa.
j) If a₁,a₂,a₃,.... and b₁,b₂,b₃,..... are two G.P.′s then a₁b₁,a₂b₂,a₃b₃ .... is also in G.P.

6. HARMONIC PROGRESSION(HP)

A sequence is said to HP if the reciprocals of its terms are in AP.
If the sequence a₁,a₂ ,a₃ ,....aₙ is an HP then 1/a₁ ,1/a₂ ,.....1/aₙ is an AP and converse. Here we do not have the formula for the sum of the n terms of an HP. The general form of a harmonic progression is
1/a,1/(a+d) ,1/(a+2d),...
1/(a+(n-1)d).

**NOTE **
If a,b ,c are in HP ⇒b=2ac/(a+c)
                          or a/c=(a-b)/(b-c).

7.    MEANS

a).    ARITHMETIC MEANS
If three terms are in AP then the middle term is called the AM between the other two, so if a ,b ,c are in AP ,. b is AM of a and c.
      n-ARITHMETIC MEANS   
      BETWEEN TWO NUMBERS .
If a,b are any two given numbers and a, A₁, A₂, ....Aₙ ,b are in AP then A₁,A₂, ...Aₙ ,are n AM′s between a and b.
A₁= a+  (b-a)/(n+1),
A₂= a+ 2(b-a)/(n+1) ,...
A₁ = a+ (b-a)/(n+1)= a+d,
   = a+2d, ...Aₙ = a+nd ,
where d = (b-a)/(n+1).

*NOTE*
Sum of n AM′s inserted beetween a and b is equal to n times the single AM between a and b i.e.
  ⁿᵣ₌₁∑ Aᵣ =nA where A is single AM between a and b.

b) GEOMETRIC MEAN

If a,b ,c are in GP , be is the GM between a and c, b²=ac,
therefore b= √(ac)
n-GEOMETRIC MEANS
between a,b
If a,b are two given numbers and a, G₁ G₂,....Gₙ , b are in GP.
Then  G₁ ,G₂ , G₃ ....... Gₙ are n GMs between a and b.
G₁ = a(b/a)¹/⁽ⁿ⁺¹⁾, =or ar
G₂=a(b/a)²/⁽ⁿ⁺¹⁾ ....... =or ar²
Gₙ= a(b/a)ⁿ/⁽ⁿ⁺¹⁾   = or arⁿ
where r= (b/a)¹/⁽ⁿ⁺¹⁾

** NOTE **
The product of n GMs between a and b is equal to nᵗʰ power of the single GM between a and b i.e. ⁿᵣ₌₁π Gᵣ = (G)ⁿ where G is the single GM between a and b.

c) HARMONIC MEAN

if a, b,c are in HP, between a and c, then b= 2ac/(a+c).

**IMPORTANT NOTE**
(i) If A,G ,H ,are respectively AM , GM , HM between two positive number a and b then
   a) G²=AH (A,G ,H constitute a
        GP) 
   b) A ≥ G ≥ H
   c) A = G = H ⇒ a = b
(ii) Let a₁ ,a₂ ,.....aₙ be n positive
     real numbers , then we define
     their arithmetic mean (A),
     geometric mean (G) and
     harmonic mean(H) as
      A = a₁ + a₂+.....+ aₙ
                          n
     G = (a₁ a₂ ......aₙ)¹/ⁿ and
     H =                         n             
           (  1/a₁ + 1/a₂ +1/a₃ + ...1/aₙ)
it can be shown that A ≥ G ≥ H. Moreover equality at either place if and only if a₁ = a₂ = ...aₙ .

8. ARITHMETICO.GEOMETRIC SERIES.

A series  each term of which is formed by multiplying the corresponding term of an AP & GP is called the Arithmetico-Geometric Series, e.g. 1+3x+5x²+7x³+ ....
Here 1,3, 5,....are in AP &
1,x,x²,x³.... are in G. P.

SUM OF N TERMS OF AN ARITHMETIC-GEOMETRIC SERIES:

Let
Sₙ
=a+(a+d)r+(a+2d)r²+..
  (a+(n-1)d)rⁿ⁻¹

then Sₙ=   a   +  dr(1-rⁿ⁻¹)
               (1-r)          (1-r)²
           - (a+(n-1)d) rⁿ , r ≠ 1
                     1- r

SUM TO INFINITY

If | r | <1 and n - ∞ then
lim ₙ→∞ rⁿ =0.
S∞ =a/(1-r)+ dr/(1-r)².

SIGMA NOTATION

a) ⁿₓ₌₁∑(aᵣ ± bᵣ)=ⁿᵣ₌₁∑aᵣ ± ⁿᵣ₌₁∑bᵣ
b) ⁿᵣ₌₁∑kaᵣ =k ⁿᵣ₌₁∑aᵣ
c) ⁿᵣ₌₁∑ k = nk . where k is constant

10.      RESULTS

a) ⁿᵣ₌₁∑ r= n(n+1)/2 (sum of the
                first n natural numbers).
b) ⁿᵣ₌₁∑ r² = n(n+1)(2n+1)/6  (sum of
     the squares of the first natural
     numbers) .
c) ⁿᵣ₌₁∑ r³ = n²(n+1)²/4=(ⁿᵣ₌₁∑r)²   
    (Sum of the cubes of the first n
     natural numbers).
d) ⁿᵣ₌₁∑r⁴ = n(n+1)(2n+1)(3n²+3n-1) e)  ⁿᵣ₌₁∑(2r - 1)=n² (sum of first n
      odd natural numbers)
f) ⁿᵣ₌₁∑ 2r = n(n+1) (sum of first n
     even natural numbers).

** NOTE**
If nᵗʰ term of a sequence is given by
Tₙ = an³+bn²+cn+d where a,b,c,d are constants, then
sum of n terms is
Sₙ = ∑ Tₙ = a∑n³ +b∑n²+c∑n+∑d
This can be evaluated using the above results.

11.    METHOD OF DIFFERENCE

If T₁T₂ ,T₃ ,.....Tₙ are the terms of a sequence then some times the terms T₂ - T₁ ,T₃ - T₂ ,..... constitute an AP/GP. nᵗʰ term of the series is determined and the sum to n terms of the sequence can easily be obtained.

** NOTE**
Remember that to find the sum of n terms of a series each term of which is composed of r factors in AP. the first factors of several terms being in the same AP, we  ′write down the nᵗʰ term, affix the next factor at the end , divide by the number of factors thus increased and by the common difference and add a constant. Determine the value of the constant by applying the initial condition ′ .   

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