Arithmetic-Geometric Series Calculator

An arithmetic-geometric series is a sequence of the form

1a¹, 2a², 3a³, 4a⁴,...

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 expression

H = 1a¹ + 2a² + 3a³... + naⁿ

If we divide this expression by a, we have

H/a = 1 + 2a¹ + 3a² + ... + 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 (aⁿ⁺¹ - 1)/(a - 1). Thus, we have

H/a = d/da [(aⁿ⁺¹ - 1)/(a - 1) - 1]

H/a = d/da [(aⁿ⁺¹ - 1)/(a - 1)]

H/a = (naⁿ⁺¹ - (n+1)aⁿ + 1)/(a - 1)²

H = (naⁿ⁺² - (n+1)aⁿ⁺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 sum

W = 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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