The sum of an infinite geometric progression can be found only if the absolute value of the common ratio (|r|) is less than 1 (|r| < 1). This condition ensures that the terms of the progression decrease in size and approach zero as the number of terms increases.
Here's the formula for calculating the sum (S) of an infinite geometric progression:
S = a / (1 - r)
where:
- a is the first term of the progression
- r is the common ratio
Example:
Consider the infinite geometric progression: 1, 1/2, 1/4, 1/8, ...
- a = 1
- r = 1/2
Applying the formula:
- S = 1 / (1 - 1/2) = 1 / (1/2) = 2
Therefore, the sum of this infinite geometric progression is 2.
Practical Insights:
- Infinite geometric progressions with |r| < 1 have a finite sum, even though they have an infinite number of terms.
- This concept has applications in various fields, including:
- Finance: Calculating the present value of a perpetuity (an annuity that continues forever).
- Physics: Analyzing the behavior of certain physical systems.
- Mathematics: Proving convergence of infinite series.
Note:
If |r| ≥ 1, the sum of the infinite geometric progression does not exist, as the terms do not approach zero and the sum diverges to infinity.
