An infinite geometric progression is a sequence of numbers where each term is found by multiplying the previous term by a constant factor called the common ratio. We can represent this sequence as:
$a, ar, ar^2, ar^3, …$
where:
- $a$ is the first term
- $r$ is the common ratio
The sum of an infinite geometric progression is the sum of all the terms in the sequence. However, not all infinite geometric progressions have a finite sum. The sum only exists if the absolute value of the common ratio is less than 1 (i.e., $|r| < 1$).
Understanding the Formula
The formula for the sum of an infinite geometric progression is:
$S = frac{a}{1 – r}$
where:
- $S$ is the sum of the infinite geometric progression
- $a$ is the first term
- $r$ is the common ratio
This formula can be derived by considering the following:
- The sum of the first $n$ terms of a geometric progression:
$S_n = frac{a(1 – r^n)}{1 – r}$
- As $n$ approaches infinity, the value of $r^n$ approaches zero:
This is true because if $|r| < 1$, then $r^n$ gets smaller and smaller as $n$ increases. For example, if $r = 0.5$, then $r^n$ becomes 0.5, 0.25, 0.125, and so on as $n$ increases. This means that as $n$ approaches infinity, $r^n$ approaches zero.
- Taking the limit as $n$ approaches infinity:
When we take the limit as $n$ approaches infinity in the formula for $S_n$, the term $r^n$ becomes zero, leaving us with:
$S = frac{a}{1 – r}$
Examples
Let’s illustrate this with some examples:
Example 1:
Find the sum of the infinite geometric progression: $1, frac{1}{2}, frac{1}{4}, frac{1}{8}, …$
Here, the first term $a = 1$ and the common ratio $r = frac{1}{2}$. Since $|r| < 1$, the sum exists. Using the formula, we get:
$S = frac{1}{1 – frac{1}{2}} = frac{1}{frac{1}{2}} = 2$
Therefore, the sum of the infinite geometric progression is 2.
Example 2:
Find the sum of the infinite geometric progression: $3, -1, frac{1}{3}, -frac{1}{9}, …$
Here, the first term $a = 3$ and the common ratio $r = -frac{1}{3}$. Since $|r| < 1$, the sum exists. Using the formula, we get:
$S = frac{3}{1 – (-frac{1}{3})} = frac{3}{frac{4}{3}} = frac{9}{4}$
Therefore, the sum of the infinite geometric progression is $frac{9}{4}$
Applications of the Formula
The formula for the sum of an infinite geometric progression has various applications in mathematics, physics, and other fields. Here are some examples:
- Calculating the value of repeating decimals: Repeating decimals can be expressed as infinite geometric progressions. For example, the repeating decimal 0.333… can be written as the infinite geometric progression: $0.3 + 0.03 + 0.003 + …$. The first term $a = 0.3$ and the common ratio $r = 0.1$. Applying the formula, we get:
$S = frac{0.3}{1 – 0.1} = frac{0.3}{0.9} = frac{1}{3}$
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Modeling exponential decay: Exponential decay processes, such as radioactive decay or the cooling of a hot object, can be modeled using infinite geometric progressions. The formula can be used to calculate the total amount of decay over an infinite period.
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Calculating the area of certain geometric figures: The formula can be used to calculate the area of certain geometric figures, such as the area of a circle or the area of a sphere.
Conclusion
The formula for the sum of an infinite geometric progression is a powerful tool that can be used to solve a variety of problems in mathematics, physics, and other fields. Understanding the formula and its derivation is essential for anyone working with infinite geometric progressions.
3. Brilliant – Infinite Geometric Series