Problem 42 - Algebra 2

Question

Find the sum of each infinite geometric series, if it exists. $$ 4-\frac{8}{3}+\frac{16}{9}+\cdots $$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{12}{5} \).

Step 1: Identify the first term and common ratio

In a geometric series, the first term is denoted by \( a \). For this series, the first term \( a \) is 4. The common ratio \( r \) is found by dividing the second term by the first term, so \( r = \frac{ \left( -\frac{8}{3} \right) }{4} = -\frac{2}{3} \).

Step 2: Check if the sum of the infinite series exists

A geometric series has a sum if the absolute value of the common ratio \( |r| < 1 \). Here, \( r = -\frac{2}{3} \) and \( |r| = \frac{2}{3} < 1 \). Therefore, the sum exists.

Step 3: Use the formula for the sum of an infinite geometric series

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

Step 4: Calculate the sum

Substitute the values of \( a \) and \( r \) into the formula: \( S = \frac{4}{1 - (-\frac{2}{3})} \). Simplify the expression: \( S = \frac{4}{1 + \frac{2}{3}} = \frac{4}{\frac{5}{3}} = 4 \times \frac{3}{5} = \frac{12}{5} \).

Key Concepts

Common Ratio

The common ratio is an essential element in understanding a geometric series. It is the factor by which each term in the series is multiplied to get the next term. To find the common ratio, you divide the second term by the first term. In the presented problem, the series is:

• First term (a): 4
• Second term: \(-\frac{8}{3}\)

To calculate the common ratio \(r\), perform \(-\frac{8}{3} \div 4\), resulting in \(-\frac{2}{3}\).
The absolute value of the common ratio, \(|r| = \frac{2}{3}\), must be less than 1 for the series to have an infinite sum. Since \(\frac{2}{3}\) is indeed less than 1, we confirm that the series is convergent. This means the sum of the infinite series can be calculated.

First Term

The first term of a geometric series is the initial number of the sequence, signifying where the series begins. It's represented by \(a\). In our example, the first term \(a\) is 4.
This initial value is crucial since it affects the formula used to calculate the sum of the series. The first term not only serves as a starting point but also determines the behavior and size of the series when combined with the common ratio. When solving problems, identifying the first term is a foundational step, as seen in the exercise presented. Remember, knowing this value is necessary for applying any formulas linked to geometric series, including calculating the sum.

Sum of a Geometric Series

The sum of a geometric series is the total of all its infinite terms, condensed down due to the convergence of the series. When the common ratio \(|r| < 1\), it implies that the series will approach a finite sum. The formula used to find the sum \(S\) of an infinite geometric series is:\[ S = \frac{a}{1-r} \]Here, \(a\) is the first term of the series, and \(r\) is the common ratio. In this exercise:

• First term \(a = 4\)
• Common ratio \(r = -\frac{2}{3}\)

We plug these values into the formula to find the sum:\[ S = \frac{4}{1 - (-\frac{2}{3})} = \frac{4}{1 + \frac{2}{3}} = \frac{4}{\frac{5}{3}} = \frac{12}{5} \]Thus, the infinite series sums up to \(\frac{12}{5}\). Understanding how to use this formula is crucial for solving similar problems, ensuring that we comprehend the nature of an infinite geometric series.