Problem 33 - Precalculus: Mathematical for Calculus

Question

The first term of a geometric sequence is \(8,\) and the second term is \(4 .\) Find the fifth term.

Step-by-Step Solution

Answer

The fifth term is \( \frac{1}{2} \).

Step 1: Identify the First Term

The first term of the sequence is given as 8. Let's denote it by \( a_1 = 8 \).

Step 2: Find the Common Ratio

The second term is given as 4, and using the formula for the second term \( a_2 = a_1 \cdot r \), where \( r \) is the common ratio, we have \( 4 = 8 \cdot r \). Solving for \( r \), we get \( r = \frac{1}{2} \).

Step 3: Calculate the Fifth Term

The fifth term of a geometric sequence can be found using the formula \( a_n = a_1 \cdot r^{n-1} \). Here, \( n = 5 \), so: \[ a_5 = 8 \cdot \left( \frac{1}{2} \right)^4 = 8 \cdot \frac{1}{16} = \frac{8}{16} = \frac{1}{2}. \]

Step 4: Validate the Calculation

Revisiting the calculation for accuracy: starting from \( a_1 = 8 \) and \( r = \frac{1}{2} \), the sequence terms are \( a_2 = 4, a_3 = 2, a_4 = 1, \) and \( a_5 = \frac{1}{2} \). This pattern confirms our calculation.

Key Concepts

Common Ratio

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant known as the "common ratio." Imagine a series of numbers where each number builds on the last by the same multiplicative factor. This factor is what we call the common ratio, commonly denoted by the symbol \( r \).
To determine the common ratio, divide any term in the sequence by the term before it. For instance, in the sequence provided, the first term is 8 and the second term is 4. Therefore, the common ratio \( r \) is calculated as:

• \( r = \frac{4}{8} = \frac{1}{2} \)

The significance of the common ratio is that it directly influences how the sequence progresses. If \( r \) is greater than 1, the sequence grows. If \( r \) is between 0 and 1, the sequence decreases, as in our example. Understanding this helps us know what kind of sequence we are dealing with.

Sequence Term Formula

The sequence term formula provides a way to find any term in a geometric sequence without listing all the previous terms. Knowing this formula is crucial in saving time and effort, especially for sequences with a large number of terms.The general rule for the \( n \)-th term of a geometric sequence is given by:

• \( a_n = a_1 \cdot r^{n-1} \)

Where:

• \( a_n \) is the \( n \)-th term
• \( a_1 \) is the first term (8 in our case)
• \( r \) is the common ratio (\( \frac{1}{2} \) here)
• \( n \) is the position of the term you want to find

For example, to find the fifth term in this sequence, you substitute \( n = 5 \) into the formula:

• \( a_5 = 8 \cdot \left( \frac{1}{2} \right)^4 = \frac{1}{2} \)

Thus, using the sequence term formula is an efficient way to calculate any particular term based on the characteristics of the sequence.

Geometric Progression

A geometric progression is a specific type of sequence where each term is a constant multiple of the previous one. This structure gives geometric sequences their characteristic exponential nature.
A geometric progression can be simple or complex depending on the common ratio. Here are a few patterns seen in geometric progressions:

• If \( r = 1 \), the sequence remains constant.
• If \( r > 1 \), each term increases, leading to exponential growth.
• If \( 0 < r < 1 \), like in our exercise, the terms decrease towards zero.

Understanding a geometric progression means recognizing the pattern of multiplication that generates the sequence. This understanding is key for solving problems like the one we've been discussing, where initially it might not seem obvious how the sequence continues.In applied mathematics, geometric progressions can model real-world phenomena like population growth or decay processes. Knowing how to identify and calculate terms in such sequences can be incredibly useful in both academic studies and practical applications.