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Sequences And Series

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Sequence and Series PDF Notes, Important Questions and Formulas

SEQUENCE

A sequence is a set of terms in a definite order with a rule for obtaining the terms.
EXAMPLE: 1, 1/2, 1/3,  , 1/n, .... is a sequence .

A sequence is a function whose domain is the set N of natural numbers. Since the domain for every sequence is the set of N natural numbers, therefore a sequence is rep-resented by its range. f: N -, R, then f(n) = tn n N is called a sequence and is denoted by
{f(1), f(2), f(3)…….} = {t1, t2, t3,....} = { tn}

 

Real Sequence

A sequence whose range is a subset of R is called a real sequence.

EXAMPLE:

  1. 2, 5, 8, 11,…………….  
  2. 4, 1, -2, -5,……………
  3. 3, -9, 27, -81, ………….

 

Types of Sequence

On the basis of the number of twins there are two types of sequence.

  1. Finite sequence: A sequence is said to be finite if it has finite number of terms.
  2. Infinite sequence: A sequence is said to be infinite if it has infinite number of twins.

 

SERIES

 By adding or subtracting the terms of a get an expression which is called a series.

 If al, a2, a3,………………, an is a sequence, then the expression

 al + a2 + a3 + ………. +an is a series.

 

EXAMPLE

(i)    1+2+3+4+ ……………..  +n

(ii)  2+4+8+16+ …………….

 

PROGRESSION

It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicit formula of the nth term. Those sequences whose terms follow certain patterns are called progressions.

 

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 - 1) d , ………… 

nth term of this AP tn = a + (n- 1)d, where d = an -  an-1 .

The sum of the first n terms of the AP is given by,

 begin mathsize 12px style straight S subscript straight n equals straight n over 2 text   end text left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket equals straight n over 2 left square bracket straight a plus divided by right square bracket. end style

Where / is the last term.

 

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 terms is constant. This constant factor is called the COMMON RATIO of the series & is obtained by dividing any term by that which immediately precedes it. Therefore a, ar, ar2, ar3, ar4,   is a GP with a as the first term & r as common ratio.

  1. nth term = a rn-1
  2. Sum of the 1st n terms i.e.begin mathsize 12px style straight S subscript straight n equals fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction comma text  if r  end text not equal to 1 end style
  3. Sum of an infinite GP when |r| < 1 when n rn 0 if |r| < 1 therefore, begin mathsize 12px style straight S subscript straight infinity equals fraction numerator straight a over denominator 1 minus straight r end fraction left parenthesis vertical line straight r vertical line text  < 1 end text right parenthesis. end style
  4. If each term of a GP be multiplied or divided by the same non-zero quantity, the resulting sequence is also a GP.
  5. Any 3 consecutive terms of a GP can be taken as a/r, a, an ; any 4 consecutive terms of a GP can be taken as air 3, air, ar, ar3 & so on.
  6.  If a ,b, c are in GP  ⇒ b2 = ac
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