The sum of an infinitely decreasing geometric progression (GP) is a finite value, calculated using a simple formula. It's a fundamental concept in mathematics with various applications in fields like finance and physics.
Understanding Geometric Progressions
A geometric progression is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio (often denoted as 'r'). An infinitely decreasing GP is one where the common ratio 'r' is between -1 and 1 (-1 < r < 1). This ensures that the terms get progressively smaller and approach zero as the number of terms increases infinitely.
The Formula
The sum (S) of an infinitely decreasing GP is given by the formula:
S = a / (1 - r)
Where:
- a is the first term of the GP.
- r is the common ratio ( |r| < 1).
This formula works because as the number of terms approaches infinity, the sum converges to a finite value.
Examples
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Example 1: Consider a GP with a = 2 and r = 1/2. The sum would be: S = 2 / (1 - 1/2) = 4.
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Example 2: A GP with a = 1 and r = -1/3. The sum would be: S = 1 / (1 - (-1/3)) = 3/4.
Practical Insights
The concept of an infinitely decreasing GP is crucial for:
- Calculating present value of perpetuities: In finance, perpetuities are investments that pay out indefinitely. The present value can be calculated using the formula for the sum of an infinitely decreasing GP.
- Understanding physical phenomena: Some physical processes can be modeled using geometric series. For instance, the decay of radioactive substances follows this pattern.
The provided references support the concept of finding the sum of an infinitely decreasing GP and highlight the use of this concept in different contexts, such as finding the greatest value of a function (as indicated in one of the references), or solving problems where sums of squares or cubes of the terms are given.
