Problem 42 - Algebra 2

Question

Find \(S_{n}\) for each geometric series described. $$ a_{2}=-36, a_{5}=972, n=7 $$

Step-by-Step Solution

Answer

The sum \(S_7\) is 6564.

Step 1: Identify the Common Ratio

Given that we know the second term \(a_2 = -36\) and the fifth term \(a_5 = 972\), we can use the formula for the general term of a geometric series: \(a_n = a_1 \, r^{n-1}\). From the second term, \(-36 = a_1 \, r\), and from the fifth term, \(972 = a_1 \, r^4\). Divide the equation for \(a_5\) by the equation for \(a_2\): \(\frac{972}{-36} = \frac{a_1\, r^4}{a_1\, r}\), simplifying to \(r^3 = -27\). Solve for \(r\) to get \(r = -3\).

Step 2: Find the First Term \(a_1\)

Using \(a_2 = -36\) and the common ratio \(r = -3\), we substitute back into the equation for \(a_2\): \(-36 = a_1 \times (-3)\). Solving for \(a_1\), we get \(a_1 = 12\).

Step 3: Use the Sum Formula for a Finite Geometric Series

The formula for the sum of the first \(n\) terms of a geometric series is \(S_n = a_1 \frac{1 - r^n}{1 - r}\). Substitute \(a_1 = 12\), \(r = -3\), and \(n = 7\) into the equation to get: \[S_7 = 12 \frac{1 - (-3)^7}{1 - (-3)}\].

Step 4: Simplify the Expression

Calculate \((-3)^7 = -2187\). Substitute back into the expression: \[S_7 = 12 \frac{1 - (-2187)}{1 + 3} = 12 \frac{1 + 2187}{4}\]. Simplify to get \[S_7 = 12 \times 547 = 6564\].

Key Concepts

Common Ratio

In a geometric series, each term after the first is obtained by multiplying the previous term by a fixed non-zero number known as the common ratio. The common ratio, often denoted by the symbol \(r\), is crucial in identifying the pattern of the series. It can be either a positive or negative number and can also take fractional values, depending on the sequence.

• If \(r > 1\), the terms will grow larger.
• If \(0 < r < 1\), the terms will get smaller.
• If \(r = 1\), the series is constant as each term is the same.
• If \(r < 0\), the series will alternate in sign.

To find the common ratio in our example, we used the relationship between two known terms: \(a_2 = -36\) and \(a_5 = 972\). Using the formula \(a_5 = a_2 \cdot r^3\), solving for \(r\) yielded \(r = -3\). This tells us that each term is obtained by multiplying the previous term by \(-3\).

General Term of a Geometric Series

The general term of a geometric series, usually denoted as \(a_n\), describes a specific term in the series as a function of its position. The formula for this is given by: \[ a_n = a_1 \cdot r^{n-1}\]where:

• \(a_1\): The first term
• \(r\): The common ratio
• \(n\): The position of the term in the series

This formula is handy when you need to find any term of the series without calculating all preceding terms. In the solved exercise, applying this formula helped link the second term \(a_2\) and the fifth term \(a_5\) through the common ratio, validating the consistent progression of the series.

Sum of Finite Geometric Series

The sum of a finite geometric series is calculated using a specific formula that provides the sum of all terms up to a certain point \(n\). This formula is expressed as:\[ S_n = a_1 \frac{1 - r^n}{1 - r} \]where:

• \(S_n\): The sum of the first \(n\) terms
• \(a_1\): The first term of the series
• \(r\): The common ratio
• \(n\): The number of terms

This formula is essential for quickly determining the sum of the series without individually adding up each term. In the exercise, this formula was used to calculate the sum \(S_7\) for our series with \(a_1 = 12\), \(r = -3\), and \(n = 7\), eventually resulting in the value \(6564\).

First Term of a Geometric Series

Identifying the first term of a geometric series, denoted by \(a_1\), is crucial as it sets the starting point of the sequence from which all other terms are derived. Often, the first term is provided directly. However, when not available, it can be deduced using any known term and the common ratio.In the given problem, the first term \(a_1\) was found using the relationship between the second term (\(a_2 = -36\)) and the common ratio \(r = -3\). By rearranging the general term formula \(a_2 = a_1 \cdot r^{2-1}\), solving for \(a_1\), we obtained \(a_1 = 12\). This first term is crucial for further calculations, such as determining the sum of the series or predicting the progression pattern.