Problem 55 - Algebra 2

Question

Evaluate the infinite geometric series \(\frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\ldots\) Enter your answer as a fraction.

Step-by-Step Solution

Answer

The sum of the infinite geometric series \(\frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\ldots\) is \(\frac{2}{3}\).

Step 1: Identify the first term and the common ratio

The first term (a) of the series is \(\frac{2}{5}\). The common ratio (r) can be found by dividing the second term by the first term, which gives us \(\frac{4}{25} / \frac{2}{5} = \frac{2}{5}\)

Step 2: Apply the geometric series sum formula

The formula for the sum of an infinite geometric series is \(S = a / (1 - r)\). Plug the values of a and r into this formula: \(S = \frac{2}{5} / (1 - \frac{2}{5})\).

Step 3: Simplify the fraction

Simplifying, we get \(S = \frac{2}{5} / \frac{3}{5} = 2/3\).

Key Concepts

Understanding the Common Ratio in Geometric Series

In an infinite geometric series, the "common ratio" is a pivotal concept. The common ratio, denoted by \(r\), is the factor by which we multiply each term to get the next one. In this context, our series looks like this: \(\frac{2}{5}, \frac{4}{25}, \frac{8}{125}, \ldots\). To find the common ratio \(r\), we need to divide the second term by the first term.

Let's do the math: Divide \(\frac{4}{25}\) by \(\frac{2}{5}\). When dividing fractions, we multiply by the reciprocal. So, \(\frac{4}{25} / \frac{2}{5}\) becomes \(\frac{4}{25} \times \frac{5}{2}\). Calculating further, we find that \(r = \frac{2}{5}\).

Thus, every term in the series is \(\frac{2}{5}\) times the previous one, a defining feature of geometric series.

The Geometric Series Sum Formula

The geometric series sum formula is essential for computing the sum of an infinite geometric series, especially when we're dealing with infinite terms. The formula is expressed as:

• \(S = \frac{a}{1 - r}\)

Here, \(S\) is the sum of the series, \(a\) is the first term, and \(r\) is the common ratio.

For our series, \(a = \frac{2}{5}\) and \(r = \frac{2}{5}\). Plugging these values into the formula gives us:

• \(S = \frac{\frac{2}{5}}{1 - \frac{2}{5}}\)
• \( 1 - \frac{2}{5} = \frac{3}{5}\)

This simplifies the expression under the formula, preparing it for further calculation. The beauty of this formula is its ability to connect a seemingly never-ending series to a finite value.

Fraction Simplification

Simplifying fractions is a key mathematical skill used to find easier numbers to work with, especially when applying the geometric series sum formula. After plugging in the variables, we have:

• \(S = \frac{\frac{2}{5}}{\frac{3}{5}}\)

When simplifying, first multiply by the reciprocal of the denominator. This transforms our expression into:

• \(\frac{2}{5} \times \frac{5}{3} = \frac{2}{3}\)

Notice that the \(5\)s cancel each other out. Multiplying what's left gives \(2\) over \(3\).

So, the sum of this infinite geometric series is \(\frac{2}{3}\). Mastering fraction simplification makes handling complex algebra much smoother and less intimidating.