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# Arithmetic-Geometric Series Calculator

An arithmetic-geometric series is a sequence of the form

1*a¹*, 2*a²*, 3*a³*, 4*a⁴*,...

for some constant *a*. This series is so-called because it resembles the geometric series

*a¹*, *a²*, *a³*, *a⁴*,...

and the arithmetic progression

1, 2, 3, 4,...

The arithmetic-geometric series is a type of hypergeometric series, i.e, a series in which the ratio of consecutive terms is a rational function of the index. Such sequences arise in discrete mathematical applications.

You can use the calculator on the left to evaluate a finite or infinite arithmetic-geometric sum.

### Formula for the Sum of an Arithmetic-Geometric Series

To find the sum of a finitely many terms in an arithmetic-geometric series, consider the expressionH = 1

*a¹*+ 2

*a²*+ 3

*a³*... +

*na*

^{n}If we divide this expression by

*a*, we have

H/

*a*= 1 + 2

*a¹*+ 3

*a²*+ ... +

*na*

^{n-1}The quantity on the right-hand side is the derivative of the geometric series summed from

*k*= 1 to

*n*. Recall that the sum of a geometric series from 0 to

*n*is (

*a*- 1)/(

^{n+1}*a*- 1). Thus, we have

H/

*a*=

*d*/

*da*[(

*a*- 1)/(

^{n+1}*a*- 1) - 1]

H/

*a*=

*d*/

*da*[(

*a*- 1)/(

^{n+1}*a*- 1)]

H/

*a*= (

*na*- (

^{n+1}*n*+1)

*a*+ 1)/(

^{n}*a*- 1)²

H = (

*na*- (

^{n+2}*n*+1)

*a*+

^{n+1}*a*)/(

*a*- 1)²

If |

*a*| is less than one, the sum converges as

*n*goes to infinity. The infinite sum is

*a*/(

*a*- 1)².

### Example

Simplify the sumW = 5 + 6(0.3) + 7(0.3)² + 8(0.3)³ + ... + 18(0.3)¹³

First, multiply both sides by (0.3)⁵ to obtain

W(0.3)⁵ = 5(0.3)⁵ + 6(0.3)⁶ + 7(0.3)⁷ + ... + 18(0.3)¹⁸

Now add 1(0.3) + 2(0.3)² + 3(0.3)³ + 4(0.3)⁴ to both sides:

W(0.3)⁵ + 1(0.3) + ... + 4(0.3)⁴ = 1(0.3) + ... + 18(0.3)¹⁸

W(0.3)⁵ + [4(0.3)⁶ - 5(0.3)⁵ + 0.3]/0.49 = [18(0.3)²⁰ - 19(0.3)¹⁹ + 0.3]/0.49

W(0.3)⁵ = [18(0.3)²⁰ - 19(0.3)¹⁹ - 4(0.3)⁶ + 5(0.3)⁵]/0.49

W = [18(0.3)²⁰ - 19(0.3)¹⁹ - 4(0.3)⁶ + 5(0.3)⁵]/[0.49(0.3)⁵]

W = [18(0.3)¹⁵ - 19(0.3)¹⁴ - 4(0.3) + 5]/0.49

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