Question

How to check if this series is in AGP(Arithmetico Geometric progression) :



1+2x+$3x^2$+$4x^3$+...


​

Answer 1

The answer is attachment due to the inappropriate word.

Answer 2

$\large\underline{\sf{Solution-}}$

The given series is

$\rm :\longmapsto\:1 + 2x + {3x}^{2} + {4x}^{3} + - - - - -$

is a product of corresponding terms of 2 series.

First series is

$\rm :\longmapsto\:1,2,3,4, - - - -$

and

Second series is

$\rm :\longmapsto\:1,x, {x}^{2} , {x}^{3} , - - - -$

Now,

Let we check the first series,

$\rm :\longmapsto\:1,2,3,4, - - - -$

Since,

$\rm :\longmapsto\:2 - 1 = 3 - 2 = 4 - 3 = - -$

So, its an AP series

Now,

Let we check the second series,

$\rm :\longmapsto\:1,x, {x}^{2} , {x}^{3} , - - - -$

Since,

$\rm :\longmapsto\:\dfrac{x}{1} = \dfrac{ {x}^{2} }{x} = \dfrac{ {x}^{3} }{ {x}^{2} } = - -$

Therefore, its an GP series.

So, the given series is a product of corresponding terms of AP and GP, therefore its an AGP series.