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:
$S = frac{0.3}{1 – 0.1} = frac{0.3}{0.9} = frac{1}{3}$
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.
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