SEQUENCE AND SERIES

INTRODUCTION:

SEQUENCE: A sequence is a function whose domain is the set N of natural numbers.

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

In other words,a real sequence is a function with domain N and the range a subset of the set R of real numbers.

SERIES:

If a1, a2, a3, a4, .... an is a sequence, then the expression 

a1+ a2+a3+a4+a5+...+an+... is a series.

PROGRESSIONS:

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

·        Every Progression is a series but every Series need not be a Progression.

·        They are 4 types of Progressions (mainly 3)

1)    Arithmetic Progression

2)    Geometric Progression

3)    Harmonic Progression

4)    Arithmetico-Geometric Progression

ARITHMETIC PROGRESSION (A.P.):

A sequence  is called an arithmetic progression if the difference of a term and the previous term is always same i.e. 

The constant difference, generally denoted by d is called the common difference.

ILLUSTRATION 1

1,4,7,10,… is an A.P. whose first term is 1 and the common difference is 4-1=3.

ILLUSTRATION 2

11 ,7,3,-1, … is an A.P. whose first term is 11 and the common difference 7-11=-4.

PROPERTIES OF AN ARITHMETIC PROGRESSION:

PROPERTY I :
If a is the first term and d the common difference of an A.P., then its nth terms an is given by

PROPERTY II:

A sequence is an A.P if its nth term is of  the form case is An+ B i.e. a linear expression in n. The common difference is such a case is A i.e. the coefficient of n.

PROPERTY III:

If a constant is added to or subtracted from each term of an A.P., then the resulting sequence is also an A.P. with the same common difference.

PROPERTY IV:

If each  term of a given A.P. is multiplied or divided by a non-zero constant k, then the resulting sequence is also an A.P. with  common difference kd or d/k , where d is the common difference of the given AP.

PROPERTY V:

In a finite A.P. the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term i.e. 

for all k = 1 ,2 ,3 ,…, n-1.

PROPERTY VI:

Three numbers a,b,c are in A.P. if 2b =a+c.

PROPERTY VII:

If the terms of an A.P. are chosen at regular intervals,  then the form of an A.P.

PROPERTY VIII:
If an, an+1 and an+2 three consecutive terms of an A.P., then.,

SELECTION OF TERMS IN AN A.P.

The following ways  of selecting terms are generally very convenient.

SOME USEFUL RESULTS

AN IMPORTANT PROPERTY:

A sequence is an A.P. if and only if the sum of its n terms is of the form 

where A, B are constants. In such a case, the common difference of the A.P is 2A.

REMARK:

It follows from this property that a sequence is an A.P. if the sum of its n terms is of the form

i.e., a quadratic expression in n and in such a case tehe common difference is twice the coefficient of n2. For example, if

 we can say theat it is sum of n terms of an A.P with common difference 6.

INSERTION OF ARITHMETIC MEANS:

• If between two given quantities a and b we have to insert n quantities A1,A2,...An, such that A1,A2,...An, b form an A.P. Then we say that A1,A2,...An are arithmetic means between a and b.
• Example: Since 15,11,7,3,-1,-5 are in A.P.,it follows that 11,7,3,-1 are four arithmetic means between 15 and -5. 

INSERTION OF n ARITHMETIC MEANS BETWEEN a AND b:

Let A1,A2,...An be n arithmetic means between a and b.
Then,a,A1,A2,...An,b is an A.P.
Let d be the commom difference of this A.P. Clearly,it contains (n+2) terms.

Now,

These are the required arithmetic means between a and b.

INSERTION OF A SINGLE ARITHMETIC MEAN BETWEEN a AND b:
Let A be the arithmetic mean of a and b. Then a,A,b are in A.P.

Laws Of ThermoDynamics.....

Hello Guys...!!
Here is a fast over-view of  Laws Of ThermoDynamics.

Zeroth Law of Thermo dynamics: Let A, B, C be different bodies having different temperatures. If heat exchange is possible between A and B and B and C then it is also possible between A and C.

Work and Heat:

      Work:

i)                    It is the mode of energy transfer mechanical work 

ii)                  It is a path dependent function

iii)                Work done on system increases the energy of a system and work done by system decreases the energy of system
Heat:

i)                    It is the quantity of heat which flows between system and surrounding.

ii)                  Always flows from higher temperature to lower temperature.

Internal Energy:

Summation of all types of energies constitutes to give net internal energy of a particular system.

First law of thermo dynamics:

            1)      Energy can neither be created nor be destroyed. It can be only transformed from one form to another. 

            2)      Total energy of the universe is constant.

                        3)   Whenever a quantity of one king of energy disappears, an exactly equivalent quantity of energy in some other form must appear.

                                                                

          

Enthalpy:

Heat content at constant pressure is called “Enthalpy”.

 Heat capacity:

Quantity of heat required to raise the temperature of the system by one degree.

Heat capacity 

Heat capacity at constant volume 

Heat capacity at constant pressure 

                                                              

Expansion of an Ideal gas:

 1.    Isothermal expansion:

a)      Work done in reversible isothermal process

b)      Work done for irreversible process –

2.      Adiabatic process:

                                      for an adiabatic process.

        Important results for adiabatic process :
                                                                  

     

  Work done for reversible adiabatic process 

               Work done for irreversible adiabatic process

Second Law of Thermodynamics –

The Second Law of Thermodynamics states that the state of entropy of the entire universe, as a closed isolated system, will always increase over time. The second law also states that the changes in the entropy in the universe can never be negative.

·         Second law of thermodynamics gives rise to the concept of ENTROPY

·         Entropy (S) - measure of the number of specific ways in which a thermodynamic system may be arranged, commonly understood as a measure of disorder.

Gibb’s Free Energy (G)

    G = H –TS

 at constant temperature.

    Also,

    At constant temperature, 

Criteria for Spontaneity:

For a spontaneous process, entropy of universe must increase, i.e.

(i)                 If and is always negative, always spontaneous.

(ii)               Ifandis always positive, always non-spontaneous

(iii)             If and, process will be spontaneous higher temperature but non-spontaneous at lower temperature.

(iv)             If andprocess will be spontaneous lower temperature but non-spontaneous at higher temperature.

. Puzzles..........

  Hello MyRankers, 

  Here Are Few Questions. Please Find Their Answers                                           

1.	Using the five letters below only, create a nine-letter word. LOPER
2.	Using the four letters below only, create a seven letter word. UMNI

3.	What is the missing letter?

Do comment your answers below to get to know the answers and explanation soon...!!

“No one in the world get succeed in the first attempt. Those who learn things from their mistakes can only lead to success”. To know read this short story.

                                                 Learn from Mistakes

                 Thomas Edison tried two thousand different materials in search of a filament for the light bulb. When none worked satisfactorily, his assistant complained, “All our work is in vain. We have learned nothing.”

                 Edison replied very confidently, “Oh, we have come a long way and we have learned a lot. We know that there are two thousand elements which we cannot use to make a good light bulb.”

Attitude says what you are…………?

THE STORY OF A BLIND GIRL

There was a blind girl who hated herself just because she was blind. She hated everyone, except her loving boyfriend. He was always there for her. She said that if she could only see the world, she would marry her boyfriend. One day, someone donated a pair of eyes to her and then she could see everything, including her boyfriend. Her boyfriend asked her, “Now that you can see the world, will you marry me?”

The girl was shocked when she saw that her boyfriend was blind too, and refused to marry him. Her boyfriend walked away in tears, and later wrote a letter to her saying:

“Just take care of my eyes dear.”

This is how human brain changes when the status changed. Only few remember what life was before, and who’s always been there even in the most painful situations.

Life Is A Gift

Today before you think of saying an unkind word – think of someone who can’t speak.

Before you complain about the taste of your food – think of someone who has nothing to eat.

Before you complain about your husband or wife – think of someone who is crying out to God for a companion.

Today before you complain about life – think of someone who went too early to heaven.

Before you complain about your children – think of someone who desires children but they’re barren.

Before you argue about your dirty house, someone didn’t clean or sweep – think of the people who are living in the streets.

Before whining about the distance you drive – think of someone who walks the same distance with their feet.

 And when you are tired and complain about your job – think of the unemployed, the

disabled and those who wished they had your job.

But before you think of pointing the finger or condemning another – remember that not one of us are without sin and we all answer to one maker.

And when depressing thoughts seem to get you down – put a smile on your face and thank God you’re alive and still around.

Life is a gift – Live it, Enjoy it, Celebrate it, and Fulfill it.