Problem 46 - Algebra 2

Question

The sum of an infinite geometric series is \(81,\) and its common ratio is \(\frac{2}{3}\) . Find the first three terms of the series.

Step-by-Step Solution

Answer

The first three terms are 27, 18, and 12.

Step 1: Identify Given Values

You are given that the sum of the infinite geometric series is \( S = 81 \) and the common ratio of the series is \( r = \frac{2}{3} \). We need to find the first three terms of the series.

Step 2: Use Infinite Geometric Series Formula

The formula for the sum of an infinite geometric series is \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Substitute the given values into this formula to find \( a \).

Step 3: Solve for the First Term

Substitute \( S = 81 \) and \( r = \frac{2}{3} \) into the formula: \( 81 = \frac{a}{1 - \frac{2}{3}} \). Simplify the denominator: \( 1 - \frac{2}{3} = \frac{1}{3} \). Thus, \( 81 = \frac{a}{\frac{1}{3}} \), which simplifies to \( a = 81 \times \frac{1}{3} = 27 \).

Step 4: Calculate the Second and Third Terms

The second term can be found by multiplying the first term by the common ratio: \( ar = 27 \times \frac{2}{3} = 18 \). The third term is found by multiplying the second term by the common ratio: \( ar^2 = 18 \times \frac{2}{3} = 12 \).

Step 5: State the First Three Terms

The first three terms of the series are: 27, 18, and 12.

Key Concepts

Geometric Series Formula

In mathematics, a geometric series is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. An infinite geometric series is a series that continues indefinitely. To determine the sum of such a series, we use the geometric series formula, which is particularly helpful when the series converges - meaning it approaches a specific value as more terms are added.

This formula for the sum of an infinite geometric series is given by:

• \( S = \frac{a}{1 - r} \)

Here, \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio. This formula works when the absolute value of \( r \), the common ratio, is less than 1 \(|r| < 1\).

This condition ensures that the terms of the series are getting smaller and that the series has a finite sum.

Sum of Series

The sum of an infinite geometric series is a fascinating aspect because it provides a way to quantify the total sum of an endless array of numbers. In the given problem, the entire sum of the series was specified as \(81\). By starting from a given sum, we can work backward using the geometric series formula to find the first term of the series.

This is done as follows:

• First, rearrange the formula: \( S = \frac{a}{1 - r} \), where \( S = 81 \) and \( r = \frac{2}{3} \).
• Substitute the known values into the equation to solve for \( a \):
  \( 81 = \frac{a}{1 - \frac{2}{3}} \).

This setup allows us to find the first term of the series, which gives the foundation for uncovering other terms in the sequence.

Common Ratio

The common ratio in a geometric series is a crucial determinant of the series' behavior. It is the consistent factor by which we multiply each term to get the next term. In our problem, the common ratio \( r \) is \( \frac{2}{3} \).

A few important things about the common ratio include:

• The common ratio can be positive or negative, but for a converging infinite series, its absolute value must be less than 1 \(|r| < 1\).
• It determines how quickly the terms of the series decrease and approach zero, which is why we can have a finite sum for an infinite series.

In the given series, each term is reduced by multiplying by \( \frac{2}{3} \), leading to progressively smaller terms and allowing the series to converge to a sum of 81.

First Term Calculation

The first term of a geometric series is pivotal as it sets the stage for the entire series. To find the first term, we used the sum formula for an infinite series. We had:

• The sum \( S = 81 \).
• The common ratio \( r = \frac{2}{3} \).
• The formula \( S = \frac{a}{1 - r} \).

Rearranging the formula and substituting in the known values gives:

• \( 81 = \frac{a}{1 - \frac{2}{3}} = \frac{a}{\frac{1}{3}} \), which simplifies to \( a = 81 \times \frac{1}{3} \).
• So, the first term \( a = 27 \).

This first term is critical as it allows you to find subsequent terms by multiplying by the common ratio, thus defining the structure of the series. Understanding this process not only highlights how each term relates to its predecessor but also emphasizes the elegance of geometric series in simplifying complex sequences.