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SEQUENCE AND SERIES:

Here we give you a complete meaning, formulae in:

1.                  A.P (2.) G.P (3.) H.P (4.) BINOMIAL SERIES (5.) EXPONENTIAL SERIES (6.) LOGARITHMIC SERIES

 

Sequence:

A sequence is a function from the set of natural numbers to the set of real numbers.

 If the sequence is denoted by the letter a, then the image of nN under the sequence a is a(n) = an.

Since the domain for e very sequence is the set of natural numbers, the images of 1, 2, 3 …n… under the sequence a are denoted by  respectively.

Here  form the sequence.

“A sequence is represented by its range”.

 

Recursive formula

 

A sequence may be described by specifying it s first few terms and a formula to determine the other terms of the sequence in terms of its preceding terms.  Such a formula is called as recursive formula.

 

Terms of a sequence:

The various numbers occurring in a sequence are called its terms.  We denote the terms of a sequence by , the subscript denote the position of the term.  The nth term is called the general term of the sequence.

 

Sequence is given by the rule:

             V1 = 1

             V2 = 1

             Vn = Vn - 1 + Vn - 2 ;

This sequence is called Fibonacci sequence.

 

Series

For a finite sequence 1, 3, 5, 7, 9 the familiar operation of addition gives the symbol 1 + 3 + 5 + 7 + 9 which has the value 25.

If we consider the infinite sequence 1, 3, 5, 7… then the symbol 1 + 3 + 5 + 7 + … has no definite value, because when we add more and more terms the value steadily increases.  1 + 3 + 5 + 7 + 9 + … is called an infinite series.  Thus a series is obtained by adding the terms of a sequence.

 

If    is an infinite sequence then  is called an infinite series.  It is denoted by.

If  = then  is called the nth partial sum of the series

Arithmetic Progression:

An arithmetic progression (abbreviated as A.P) is a sequence of numbers in which each term, except the first, is obtained by adding a fixed number to the immediately preceding term.  This fixed number is called the common difference, which is generally denoted by d.

 

Arithmetic series:

The series whose terms are in A.P is called an arithmetic series.

For example, 1 + 3 + 5 + 7 + … is an arithmetic series.

 

Geometric progression:

A geometric progression (abbreviated as G.P.) is a sequence of numbers in which the first term is non-zero and each term, except the first is obtained by multiplying the term immediately preceding it by a fixed non-zero number.  This fixed number is called the common ratio and it is denoted by the letter ‘r’.

The general form of a G.P. is  with a0 and r 0, the first term is ‘a’.

 

Geometric series:

The series  is called a geometric series because the terms of the series are in G.P.  Note that the geometric series is finite or infinite according as the corresponding G.P. consists of finite (or) infinite number of terms.

 

Harmonic progression:

A sequence of non-zero numbers is said to be in harmonic progression (abbreviated as H.P.) if their reciprocals are in A.P.

The general form of H.P. is  where a0 .

nth  term of H.P. is

 

Note: there is no general formula for the sum to n terms of a H.P. as we have for A.P. and G.P.

 

Means of Progressions

 

Arithmetic mean

A is called the arithmetic mean of the numbers a and b if and only if a, A, b are in A.P.  If A is the A.M between a and b then a, A, b are in A.P.

             

 
 


                                    

 

 

A1, A2…, An are called n arithmetic means between two given numbers a and b if and only if a, A1, A2…, An , b are in A.P.

 

Geometric Mean:

G is called the geometric mean of the numbers a and b if and only if a, G, b are in G.P.

                   

                        

Note:

  1. If a and b are positive then
  2. If a and b are negative
  3. If a and b are opposite sign then their G.M is not real and it is discarded since we are dealing with real sequences.

i.e. If a and b are opposite in signs, then G.M between them does not exist.

 

Harmonic mean:

H is called the harmonic mean between a and b if a, H, b are in H.P

If a, H, b are in H.P then  are in A.P

      

 
 

 

 

 

 


This H is single H.M between a and b.

 

Definition:

H1, H2…Hn are called n harmonic means between a and b if a, H1, H2…Hn , b are in H.P.

 

Relation between  A.M, G.M and H.M.

Example: If a, b are two different positive numbers then

(i) A.M. G.M, H.M are in G.P  (ii) A.M>G.M>H.M

Some special types of series:

 

Binomial series:

 

Binomial Theorem for a Rational Index:

In the previous chapter we have already seen the Binomial expansion for a positive integral index n.  (Power is a positive integer)

A particular form is

When n is a positive integer the number of terms in the expansion is (n + 1) and so the series is a finite series.  But when it is not a positive integer, the series does not terminate and it is an infinite series.

 

Theorem:

For any rational number n other than positive integer

Provided.

Here we require the condition that should be less than 1.

To see this, put x = 1 and n = -1 in the above formula for (1 + x)n

The left side of the formula = (1 + 1)-1 =,

While the right side =  = 1 – 1 + 1 – 1 + …

Thus the two sides are not equal.  This is because, x = 1 doesn’t satisfy<1.

This extra condition <1 is unnecessary, if n is a positive integer.

Differences between the Binomial theorem for a positive integral index and for a rational index:

  1. If nN, then (1 + x)n is defined for all values of x and if n is a rational number other than the natural number, then (1 + x)n is defined only when <1.
  2. If nN, then the expansion of (1 + x)n  contains only n + 1  terms.  If n is a rational number other than natural number, then the expansion of (1 + x)n contains infinitely many terms.

 

Some particular expansions

We know that, when n is a rational index,                       (1)

Replacing x by –x, we get

                      (2)

Replacing n by –n in (1) we get 

                    (3)

Replacing x by –x in (3), we get

                    (4)

Note:

  1. If the exponent is negative then the value of the factors in the numerators are increasing uniformly by 1.
  2. If the exponent is positive then the value of the factors in the numerators are decreasing uniformly by 1.
  3. If the sign of x and n are same then all the terms in the expansion are positive.
  4. If the sign of x and n are different, then all the terms alternate in sign.

 

Special cases:

 

General term:

For a rational number n and<1, we have

 

In this expansion First term T1 = T0+1 = 1

Second term T2 =

Third term

Fourth term

The general term is

 

Exponential series

 

Exponential Theorem:

For all real values of x,

But

For all real values of x,

Thus we have the following results:

 

Logarithmic Series:

If -1 < x 1 then

This series is called the logarithmic series.

The other forms of logarithmic series are as follows:

 

Here is a site to see how progressions work:

http://www.goldenkstar.com/progression-arithmetic-geometric-software-mathematics.htm .Click and enjoy. Click Back buttom to re visit ganithguru.

 

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