The sum of the infinite terms of an Arithmetico-Geometric Progression (AGP) is given by the formula: S∞ = a / (1 - r) + dr / (1 - r)2, where |r| < 1.
Here's a breakdown of the formula and its components:
- S∞: Represents the sum of the infinite terms of the AGP.
- a: Represents the first term of the arithmetic sequence.
- d: Represents the common difference of the arithmetic sequence.
- r: Represents the common ratio of the geometric sequence.
- |r| < 1: This condition is crucial. The absolute value of the common ratio 'r' must be less than 1 for the infinite sum to converge to a finite value. If |r| ≥ 1, the series diverges and does not have a finite sum.
Explanation of the Formula:
The Arithmetico-Geometric Progression (AGP) combines an arithmetic progression (AP) and a geometric progression (GP). A typical AGP looks like this:
a, (a+d)r, (a+2d)r2, (a+3d)r3, ...
Where:
- The arithmetic part is: a, a+d, a+2d, a+3d, ...
- The geometric part is: 1, r, r2, r3, ...
The formula for the sum to infinity arises from manipulating the series and using the formula for the sum to infinity of a standard geometric progression. The derivation involves subtracting r times the series from the original series, which allows us to isolate and solve for S∞.
Example:
Consider the AGP: 2, 6/2, 10/4, 14/8, ...
Here:
- a = 2 (first term of the arithmetic sequence)
- d = 4 (common difference of the arithmetic sequence: 6-2 = 10-6 = 4)
- r = 1/2 (common ratio of the geometric sequence)
Since |r| = |1/2| < 1, we can use the formula:
S∞ = 2 / (1 - 1/2) + (4 * (1/2)) / (1 - 1/2)2
S∞ = 2 / (1/2) + 2 / (1/4)
S∞ = 4 + 8
S∞ = 12
Therefore, the sum of the infinite terms of the AGP 2, 6/2, 10/4, 14/8, ... is 12.
In summary, the sum to infinity of an AGP is a well-defined formula that relies on the first term and common difference of the arithmetic sequence, as well as the common ratio of the geometric sequence, provided that the absolute value of the common ratio is less than 1.
