Sequence and series-5

 

Arithmetico-geometric Progression(A.G.P.)

Definition:

·The combination of arithmetic and geometric progression is called arithmetico-geometric progression.

nth term of A.G.P.

·                     If a1,a2, a3, a4,.....an is an A.P.

b1, b2, b3....bn is a G.P.,

then the sequence  a1b1, a2b2, a3b3,.......anbn is said to be an arithmetico-geometric sequence.

Thus, the general form of an arithmetico geometric sequence is ;

a, (a+d)r, (a+2d)r2, (a+3d)r3.................a + (n-1)dr(n-1)

 

Sum of A.G.P.

(1)           Sum of n terms:

 The sum of n terms of an arithmetico-geometric sequence a, (a+d)r, (a+2d)r2, (a+3d)r3.................a + (n-1)dr(n-1)

arithmetic geometric sequence
 

Sum of infinite sequence:

Let |r|< 1. Then rn,rn-1 as n ®¥  and it can also be shown that   n rn ®0  as n ®¥. So, we obtain that

sum of infinite sequence

In other words, when |r|< 1 the sum to infinity of an arithmetico-geometric series is ;

sum of infinite gp

Method for Finding Sum

·                     This method is applicable for both sum of n terms and sum of infinite number of terms.

·                     First suppose that sum of the series is S, then multiply it by common ratio of the G.P. and subtract. In this way, we shall get a G.P., whose sum can be easily obtained.

Method of Difference

·                     If the differences of the successive terms of a series are in A.P. or G.P., we can find nth term of the series by the following steps :

Step I:    Denote the nth term by Tn and the sum of the series upto n terms by Sn.

Step II:   Rewrite the given series with each term shifted by one place to the right.

Step III: By subtracting the later series from the former, find Tn.

Step IV: From TnScan be found by appropriate summation.

Example:

·                     Consider the series 1+ 3 + 6 + 10 + 15 +…..to n terms.

Here differences between the successive terms are 3-1,6 – 3, 10 – 6, 15 – 10, …….i.e.,  2, 3, 4, 5,…… which are in A.P. This difference could be in G.P. also. Now let us find its sum

S= 1+3+6 +10 ...........Tn-1 +Tn

S=      1+3 +6 +10..................Tn-1 +Tn

Subtracting, we get ; 

      0 = 1 +2 + 3 + 4..........( Tn-Tn-1)-Tn

       T1 +2 + 3 + 4..........up to n terms

series

Special Series :

some special series

 

Properties of Arithmetic, Geometric, Harmonic means between two given Numbers

· Let A, G and H be arithmetic, geometric and harmonic means of two numbers a and b

Then A = (a+b)/2, G =Öab and H = 2ab/ ( a+ b)

These three means possess the following properties:

³ ³ H    Note that the equality holds only when a = b.

A, G, and H forms a geometric progression such that G2 = A x H

The equation having a and b as its roots is  =   x2-2Ax+ G2 = 0 0r  x-(a+b)x + ab = 0

The roots a, b are given by : A ± ÖA2-G2

If A, G, H are arithmetic, geometric and harmonic means between three given numbers a, b and c, then the equation having a, b, c as its roots is ;

x3- 3Ax2 + 3 G3 x/H - G3=0

Relation between A.P., G.P. and H.P.

              (1) If number of terms of any A.P./G.P./H.P. is odd, then A.M./G.M./H.M. of first and last terms is middle term of series.

(2) If number of terms of any A.P./G.P./H.P. is even, then A.M./G.M./H.M. of middle two terms is A.M./G.M./H.M. of first and last terms respectively.

(3)  If pth, qth and rth terms of a G.P. are in G.P. Then p, q, r are in A.P.

(4)  If a, b, c are in A.P. as well as in G.P. then a=b=c

(5)   If a, b, c are in A.P., then  xa, xb, xc will be in G.P.  x ¹± 1.

 

properties of AP, GP

Gp concepts and formulas

gp, ap, formulas