An arithmetico-geometric series is a sequence of numbers that combines two parts:
Example: If the arithmetic part is 1, 2, 3, 4... and the geometric part is multiplied by 1/2 each time, the series looks like this:
1 × 1/2 + 2 × 1/4 + 3 × 1/8 + 4 × 1/16 + ...
The arithmetico-geometric series has three main components:
The general form of an arithmetico-geometric series is:
S = a1r0 + a2r1 + a3r2 + ...
Here:
a1, a2, a3... are terms of the arithmetic sequence.r is the common ratio of the geometric sequence.Let’s solve an example to understand how the sum is calculated:
Problem: Find the sum of the series:
S = 1 × 1/2 + 2 × 1/4 + 3 × 1/8 + ...
S = Σ (n × rn-1), where n is the arithmetic term, and rn-1 is the geometric term.The series converges if the geometric ratio r satisfies:
|r| < 1
This means the ratio must be less than 1 in absolute value for the sum to have a finite value.
Among arithmetic and geometric sequences, there are distinct differences as well as similarities. By adding, arithmetic sequences or series track terms. Multiplications follow terms in geometric series or sequences. The similarities between AP and GP are that they both follow a set of rules. This pattern is unbreakable.
The huge benefit of AP is that it may be used to forecast future statistics and trends. By evaluating the nth term in the AP sequence, we can rapidly obtain the following figures.
To calculate a GP's typical proportion, divide the second term of the arrangement by the first term, or track down the proportion of any two consecutive terms by using the previous term as the denominator