Arithmetico-Geometric Progression
Definition
Suppose
a1, a2, a3........is an A.P. and b1,
b2, b3.......is a G.P. . Then the Progression a1b1,
a2b2, ....... is said to be an arithmetico-geometric
progression (A.G.P). Hence an arithmetico-geometric progression is of the form
ab, (a+d)br, (a+2d)br2, (a+3d)br3,.........
Sum of n Terms:
The sum Sn
of first n terms of an A.G. P. is obtained in the following way :.
Sn = ab +
(a + d)br + (a + 2d)br2 +.........+(a + (n - 2)d)brn-2 +
(a + (n - 1)d)brn-1.
Multiply both sides by r, so that
r Sn = abr + (a + d)br2+.........+(a
+ (n - 3)d)brn-2 + (a + (n - 2)d)brn-1 + (a + (n - 1)d)brn.
Subtracting, we get
(1 - r)Sn = ab + dbr + dbr2
+.......+dbrn - 2 + dbrn - 1 - (a + (n - 1)d)brn.
= 
&⇒
If -1 < r < 1, the sum of the
infinite number of terms of the progression is
= 
.
Example 1
Find the sum of the series 1.2 + 2.22
+ 3.22 + ….. + 100.2100..
Solution:
S = 1.2 + 2.22
+ 3.23 + …. + 100.2100.
2S = 1.22
+ 2.23 + …. + 99.2100 + 100.2101.
&⇒ –S = 1.2 + 1.22 + 1.23
+ …. + 1.2100 – 100.2101.
&⇒ -S = 1.2
- 100.2101
&⇒
S = -2101 + 2 + 100.2101 = 99.2101 + 2.
