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.