Problem 50 - Algebra and Trigonometry

Question

Find the sum of the infinite geometric series. $$ \frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\cdots $$

Step-by-Step Solution

Answer

The sum of the series is \(\frac{1}{2}\).

Step 1: Identify the First Term

In a geometric series, the first term is usually denoted as \(a\). For this series, the first term \(a\) is \(\frac{2}{5}\).

Step 2: Find the Common Ratio

The common ratio \(r\) of a geometric series can be found by dividing the second term by the first term. Thus, \(r = \frac{4}{25} \div \frac{2}{5} = \frac{4}{25} \times \frac{5}{2} = \frac{4 \times 5}{25 \times 2} = \frac{1}{5}\).

Step 3: Verify the Common Ratio

To ensure correctness, check further division of the terms: \(\frac{8}{125} \div \frac{4}{25} = \frac{8 \times 25}{125 \times 4} = \frac{1}{5}\). This confirms that the common ratio \(r = \frac{1}{5}\).

Step 4: Use the Infinite Sum Formula

For an infinite geometric series, the sum \(S\) is given by \(S = \frac{a}{1 - r}\), where \(|r| < 1\). Here, \(a = \frac{2}{5}\) and \(r = \frac{1}{5}\).

Step 5: Calculate the Sum

Substitute the values into the formula: \(S = \frac{\frac{2}{5}}{1 - \frac{1}{5}} = \frac{\frac{2}{5}}{\frac{4}{5}}\).

Step 6: Simplify the Expression

Simplifying the fraction, \(S = \frac{2}{5} \times \frac{5}{4} = \frac{2 \times 5}{5 \times 4} = \frac{2}{4} = \frac{1}{2}\). Thus, the sum of the series is \(\frac{1}{2}\).

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This characteristic makes geometric sequences distinct from arithmetic sequences, where each term is found by adding a fixed number to the previous term.
In the sequence given in the problem, the numbers are \( \frac{2}{5}, \frac{4}{25}, \frac{8}{125}, \ldots \). Each of these numbers is connected to the ones before and after it by a common multiplicative factor, creating a consistent pattern that continues indefinitely. Understanding how these sequences are structured helps in finding other relevant properties like their sum.

Common Ratio

The common ratio is a key part of identifying and working with a geometric sequence. It is determined by dividing any term in the sequence by the term that directly precedes it. For the sequence in the exercise, starting from the second term:

• \( r = \frac{4}{25} \div \frac{2}{5} = \frac{1}{5} \)

This fixed ratio ensures the sequence's uniformity and is crucial when calculating sums or exploring other properties. In any geometric sequence, verifying the common ratio by checking multiple terms confirms consistency. This concept is not only central to solving the problem at hand, but also to understanding how such sequences behave generally.

Sum of Series

The sum of a series is the total when all terms of the sequence are added together. In finite geometric series, this is straightforward, as only a specific number of terms need to be added. However, in an infinite geometric series, things become more complex, yet intriguing.
For sequences that are infinite, calculating the sum involves understanding the notion of convergence. Specifically, if the absolute value of the common ratio \(|r|\) is less than 1, the terms get progressively smaller, and their sum converges to a finite number. This property of infinite geometric series provides convenience, allowing calculation without needing to sum infinitely many terms.

Infinite Series Formula

The formula for finding the sum of an infinite geometric series is a powerful tool in mathematics. It states that the sum \( S \) of an infinite series is given by: \[ S = \frac{a}{1 - r} \]where \( a \) is the first term, and \( r \) is the common ratio, with \(|r| < 1\). This condition ensures that the terms become infinitesimally small, allowing the series to have a finite sum.
In our exercise, \( a \) is \( \frac{2}{5} \) and \( r \) is \( \frac{1}{5} \), fitting the formula as \( |r| < 1 \). By substituting into the formula, you get:

• \( S = \frac{\frac{2}{5}}{1 - \frac{1}{5}} = \frac{1}{2} \)

This simple, elegant formula opens up the ability to find sums in otherwise complex infinite series, showcasing the delicacy and beauty of mathematical structure.