Arithmetico-geometric progression:

Consider a₁, a₂, a₃......is an Arithmetic progression. and b₁, b₂, b₃..... is a G.P. Then the series a₁b₁, a₂b₂, ....... is called to be an arithmetico-geometric progression (A.G.P). Consequently an arithmetico-geometric progression is of the one of the form ab, (a+d)br, (a+2d)br², (a+3d)br³,.........

Summation of n Terms:

The addition S_n of first  n  terms  of an  A.G. P.  is calculated in  the  given  way :

S_n = ab + (a + d)br + (a + 2d)br² +.........+(a + (n - 2)d)br^n-2 + (a + (n - 1)d)br^n-1

Multiply each sides by r, so that

r S_n = abr + (a + d)br²+.........+(a + (n - 3)d)br^n-2 + (a + (n - 2)d)br^n-1 + (a + (n - 1)d)brⁿ

Subtracting, we obtain

(1 - r)S_n = ab + dbr + dbr² +.......+dbr^{n - 2} + dbr^{n - 1} - (a + (n - 1)d)brⁿ

If -1 < r < 1, the summation of the infinite number of terms of the progression is

Problem: Calculate the sum of the series 1.2 + 2.2² + 3.2³ + ..... + 100.2¹⁰⁰.

Solution:             => = 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

                              = -2¹⁰¹ + 2 + 100.2¹⁰¹  = 99.2¹⁰¹ + 2.

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