Arithmatic progression

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05 ARITHMATIC PROGRESSION

Arithmetic Progression:

An arithmetic progression is list of numbers in which the difference between any two consecutive terms is constant.

a1,a2 ,a3,a4 ........are in A.P. if    
 a2a1=d=a4a3=a5a4.....d = common difference  a=a1  First term.

e.g   4,710,13,……   is A. P with    a=4      and   d=3

Each member of the sequence is called term.

  • Common difference d can be zero, positive or negative.
  • General form of AP is: a,a+d,a+2d
  • A.P. can be finite or infinite.

General Term of A.P. THEOREM Let ‘a ’ be the first term and ‘d ’ be the common difference

of an A.P. Then, its nth  term or general term is given by  an=a+(n1)d

PROOF

Let  a1,a2 ,.....an be the given A.P. Thena1=a,  
a1=a+(11)d......(1)

Since each term of an A.P. is obtained by adding common difference to the preceding term. Therefore,

a2 = a+d
a2=a+(21)d

………(2)

Similarly, we have

a3=a2+d
a3=(a+d) +d
 a3 =a+2d
a3=a+(31)d
a4=(a+2d)+d......(3)
a4=a+3d
a4=a+(41)d....(4)

Observing the pattern in equation (1), 2), (3) and (4), we find that

an=a+(n1)d
nth TERM OF AN A.P. FROM THE END

Let there be an A.P. with first term ‘a ’ and common difference ‘d ’. If there are ‘m ’ terms in

the A.P., then  nth term from the end =(mn+1)th  term from the beginning
nth   term from the end =amn+1
 nth  term from the end 
=a+(mn+11)d
 nth  term from the end =a+(mn)d

Also, if I is the last term of the A.P., then  term from the end is the nth term of an A.P. whose first term is I and common difference is –d.

nth   term from the end =  Last term +(n1) d
  nth  term from the end = I(n1)d

MIDDLE TERM(S) OF A FINITE A.P.

Let there be a finite A.P. with ‘ ’n terms whose first term is ‘a ’ and common difference is ‘d’.

If n is odd, then

n+12th

term is the middle term and is given by

a+n+121d

If n is even, then

n2th andn2+1th

are middle term given by

a+n21d
and  a+n2+11d=a+n2d

respectively

Sum of n terms of AP:

Sum of first n terms of A.P. is given by

sn =n22a+n1d
or sn =n2a+an
or sn =n2a+1       (1=  last term of A.P.)
The  nth
 term also can be given by  an=snsn1

THEOREM The sum of n terms of an A.P. with first term ‘ a’ and common difference ‘ d’ is

sn=n22a+(n1)d
or     sn=n2a+a+(n1)d
or    sn =n2a+1
, where 1=   last term   =a+(n1)d
PROOF:  Let a1,a2 ,a3 be an A.P. with first term a  and common difference d . Then,
a1=a, a2=a+d ,a3= a+2d,a4=a+3d,........,an=a+(n1)d
Now , sn =a1+a2+a3+........+an
 sn =a+(a+d)+(a+2d)+.......+(a+(n2)d)+{a+(n1)d}  ......(1)

Writing the above series in a reverse order, we get

sn ={a+(n1)d}+{a+(n2)d}+.......+(a+d)+a.........(2)

Adding the corresponding terms of equations (1) and (2), we get

2sn={2a+(n1)d}+{2a+(n1)d}+......+{2a+(n1)d}
2sn= n{2a+(n1)d} [ 2a+(n1) d repeats n times]
 sn=n2{2a+(n1)d}

Now,

 I= last term =nTH  term  =a+(n1)d
 sn =n2{2a+(n1)d} =n2[a+{a +(n1)d}] =n2{a+1}

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