Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{3}=0.4, r=2, n=12 $$

Step-by-Step Solution

Answer

The sum of the first 12 terms of the geometric series is 409.5.

Step 1: Understand the Formula for Sum of a Geometric Series

The sum of the first \(n\) terms \(S_n\) of a geometric series can be given by the formula: \[ S_n = a \frac{(r^n - 1)}{r - 1} \]where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 2: Identify the First Term and Common Ratio

We know the common ratio \(r = 2\) and the term \(a_3 = 0.4\) (which is the third term in the sequence). We need to determine the first term \(a\) of the geometric series.

Step 3: Calculate the First Term

To find the first term \(a\), use the formula for the nth term of a geometric sequence:\[ a_n = a \, r^{n-1} \]For the third term, \(a_3 = a \, r^2\).Thus, \[ 0.4 = a \, (2)^2 \]\[ 0.4 = 4a \]\[ a = \frac{0.4}{4} = 0.1 \]

Step 4: Substitute Values into the Formula

We now substitute the values \(a = 0.1\), \(r = 2\), and \(n = 12\) into the sum formula:\[ S_{12} = 0.1 \frac{(2^{12} - 1)}{2 - 1} \]

Step 5: Compute the Sum

First, compute \(2^{12}\): \[ 2^{12} = 4096 \]Then, calculate:\[ S_{12} = 0.1 \times (4096 - 1) \]\[ S_{12} = 0.1 \times 4095 = 409.5 \]

Key Concepts

Sum of Geometric Series

In a geometric series, each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. The sum of the series is essentially the total of all these terms added together. When you're given the task to find the sum of the first \( n \) terms, there's a straightforward formula that helps simplify this process:

• \( S_n = a \frac{(r^n - 1)}{r - 1} \)

Here, \( S_n \) represents the sum of the first \( n \) terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the total number of terms we want the sum for. By substituting the values of \( a \), \( r \), and \( n \) into this formula, like in our example, you can easily calculate the sum. Understanding this formula allows you to efficiently find the sum without manually adding each term.

Common Ratio

The common ratio is a key component of any geometric series and dictates how each term in the series is related to its predecessor. It is defined as the constant factor that you multiply by one term to get the next. In our example, the common ratio \( r \) is 2.

• If \( r = 2 \), each term is twice the previous term.
• If \( r \) is greater than 1, the terms increase.
• If \( r \) is between 0 and 1, the terms decrease.

Understanding the common ratio is crucial because it not only helps in identifying the pattern in the series but also plays a role in the formula for the sum of the series. By understanding the nature of \( r \), you can predict whether the series will grow or shrink as you move through terms.

First Term of Geometric Series

The first term of a geometric series, denoted as \( a \), is the initial value in the sequence from which all other terms are derived. It's fundamental to know this because all subsequent terms are generated from it using the common ratio.

• In our example, we calculated the first term as \( a = 0.1 \).

Finding the first term can sometimes require solving an equation, particularly if you're given a specific term rather than the first. Using the formula for the nth term, \( a_n = a \cdot r^{n-1} \), you can rewrite this to solve for \( a \) when you know another term in the series, as shown in the step-by-step solution.

Nth Term Formula

To find any given term in a geometric series, including perhaps finding the first term from a known nth term, you use the nth term formula:

• \( a_n = a \cdot r^{n-1} \)

This formula shows how terms are exponentially generated from the first term based on its position (\( n \)) in the series and the common ratio (\( r \)). When you rearrange this formula, it becomes a vital tool for identifying unknown variables within a series.
It's also helpful for verifying given terms' calculations within your series. If you know one of the terms and the common ratio, you can determine the first term by solving for \( a \), ensuring your series follows the correct pattern.