An arithmetico-geometric series is the sum of consecutive terms in an arithmetico-geometric sequence defined as: , where and are the th terms of arithmetic and geometric sequences, respectively.
The sum of the first n terms of an is , where is the common difference of and is the common ratio of . Or, , where is the sum of the first terms of .
Let represent the sum of the first n terms.
The sum of an infinite arithmetico-geometric sequence is , where is the common difference of and is the common ratio of (). Or, , where is the infinite sum of the .