Problem 43 - Algebra 2

Question

Critical Thinking Find the specified value for each infinite geometric series. $$ a_{1}=12, S=96 ; \text { find } r $$

Step-by-Step Solution

Answer

The common ratio \( r \) for the infinite geometric series is \( 7/8 \).

Step 1: Understanding the Problem

The problem tells us that we have an infinite geometric series, where the first term \( a_{1} = 12 \), the sum \( S = 96 \), and we need to solve for the common ratio \( r \).

Step 2: Use the formula for the Sum of an Infinite Geometric Series

The formula for the Sum of an Infinite Geometric Series is \( S = a_1 / (1 - r) \). This formula describes the sum \( S \) of a geometric series as its first term divided by (1 minus the common ratio).

Step 3: Rearrange the formula and substitute to solve for r

Rearrange the formula to solve for \( r \). It will look like this: \( r = 1 - a_1 / S \). Now we substitute the given values \( a_{1} = 12 \) and \( S = 96 \) and solve for \( r \). This gives us \( r = 1 - 12 / 96 = 1 - 1/8 = 7/8 \).

Key Concepts

Geometric Series Sum

An infinite geometric series 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. The series goes on forever, and its sum can be found using a special formula. The sum of an infinite geometric series is given by:\[S = \frac{a_1}{1 - r}\]where:

• \( S \): represents the sum of the series
• \( a_1 \): is the first term of the series
• \( r \): is the common ratio of the series

This formula only works if the absolute value of the common ratio \( r \) is less than 1. If \( |r| \geq 1 \), the series does not converge, meaning it doesn't have a finite sum. The beauty of this formula is that it allows us to find the sum of an infinite number of terms in the series quickly, without having to add them manually. In the given problem, since the sum \( S = 96 \), and the first term \( a_1 = 12 \), we can use the formula to find the common ratio \( r \) of the series.

Common Ratio

The common ratio in a geometric series is a crucial element that determines the progression of terms. It is the factor by which each term is multiplied to get the next term. Mathematically, the common ratio \( r \) can be calculated if you have consecutive terms \( a_{n+1} \) and \( a_n \), using the expression:\[r = \frac{a_{n+1}}{a_n}\]In the context of infinite geometric series, ensuring the common ratio satisfies \( |r| < 1 \) is necessary for the series to have a finite sum, thus confirming its convergence. In the given problem, understanding that the series sum formula \( S = \frac{a_1}{1 - r} \) can also be rewritten to solve for \( r \):\[ r = 1 - \frac{a_1}{S}\]Using this rearranged formula helps in calculating \( r \) when the first term and the series sum are known. By substituting the values provided, \( a_1 = 12 \) and \( S = 96 \), the calculation yields:\[ r = 1 - \frac{12}{96} = 1 - \frac{1}{8} = \frac{7}{8}\]As \( \frac{7}{8} \) is indeed less than 1, we confirm that this series converges, reflecting the importance of the common ratio in the analysis of geometric series.

First Term

The first term of a geometric series, often denoted as \( a_1 \), is the starting point of the sequence. It plays a significant role in calculating the total sum of the series. This term is not only essential in initiating the sequence but also critical in determining the sum when used in conjunction with the common ratio.In the formula:\[S = \frac{a_1}{1 - r}\]The first term \( a_1 \) helps determine how much of the sum is contributed right from the start before the effects of the common ratio take full effect. In solving for any given value in the context of an infinite geometric series, knowing the first term allows us to apply the correct calculation to find either the sum or another missing value like the common ratio.For the exercise presented, knowing that \( a_1 = 12 \) provided a foundation for employing the sum formula and ultimately solving for the common ratio \( r \). By highlighting the role of \( a_1 \), one can better appreciate the mechanics of how an infinite geometric series behaves when calculating its sum.