·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)
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
In other words, when |r|< 1 the sum to infinity of an arithmetico-geometric series is ;
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.
· 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, Sn can be found by appropriate summation.
· 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
Tn = 1 +2 + 3 + 4..........up to n terms
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:
A ³ G 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 x2 -(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.