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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.

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 Tn, Scan 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

    T= 1 + 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:

A³ G³ H    Note that the equality holds only when a = b.

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

properties of AP, GP

Gp concepts and formulas

gp, ap, formulas