Problem 28 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ -8,-2,-\frac{1}{2},-\frac{1}{8}, \dots $$

Step-by-Step Solution

Answer

The common ratio is \( \frac{1}{4} \), the fifth term is \( -\frac{1}{32} \), and the nth term is \( a_n = -8 \cdot \left(\frac{1}{4}\right)^{n-1} \).

Step 1: Identify the First Term

The first term of the given geometric sequence is the first number in the list: \( a_1 = -8 \).

Step 2: Determine the Common Ratio

The common ratio \( r \) of a geometric sequence is found by dividing the second term by the first term. Using the terms \(-8\) and \(-2\), calculate \( r = \frac{-2}{-8} = \frac{1}{4} \).

Step 3: Calculate the Fifth Term

The formula for the \(n\)th term of a geometric sequence is \( a_n = a_1 \cdot r^{n-1} \). To find the fifth term, substitute \( n=5 \), \( a_1 = -8 \), and \( r = \frac{1}{4} \): \[a_5 = -8 \cdot \left(\frac{1}{4}\right)^{4} = -8 \cdot \frac{1}{256} = -\frac{1}{32}.\]

Step 4: Write the nth Term Formula

Substitute \( a_1 = -8 \) and \( r = \frac{1}{4} \) into the nth term formula \( a_n = a_1 \cdot r^{n-1} \):\[ a_n = -8 \cdot \left(\frac{1}{4}\right)^{n-1}. \]

Key Concepts

Common Ratio

The common ratio in a geometric sequence is an important factor that helps determine the relationship between successive terms. In simple terms, it is the number you multiply by each term to get the next one.
In our example sequence,

• First term (\(-8\))
• Second term (\(-2\))

To find the common ratio (\(r\)), divide the second term by the first term: \(r = \frac{-2}{-8} = \frac{1}{4}\).
This means each term is one-fourth of the term before it, indicating the sequence is decreasing.

Nth Term

Figuring out the nth term in a geometric sequence allows you to calculate the value of any term in the sequence without listing all preceding ones. This is especially helpful for large values of \(n\).
The formula for the nth term is: \(a_n = a_1 \cdot r^{n-1}\). Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number you are calculating for.
To use the formula effectively:

• Substitute the first term (\(-8\)) and the ratio (\(\frac{1}{4}\))
• Plug in the value of \(n\)

This gives you \(a_n = -8 \cdot \left(\frac{1}{4}\right)^{n-1}\), letting you calculate any term easily.

First Term

The first term of a geometric sequence is crucial as it serves as the starting point for finding any other terms in the series. In our sequence \(-8, -2, -\frac{1}{2}, -\frac{1}{8}, \dots \), the first term is \(a_1 = -8\).
When using the nth term formula, the first term defined as \(a_1\) plays a pivotal role. The sequence builds on this number using the common ratio.
In practical terms, knowing the first term provides a concrete foundation, allowing you to understand the sequence better right from the beginning.

Fifth Term

The fifth term is a specific instance of understanding how to use the nth term formula. Calculating the fifth term showcases how patterns unfold throughout the sequence. Using our formula \(a_n = a_1 \cdot r^{n-1} \) and substituting \(n = 5\):

• First term (\(a_1 = -8\))
• Common ratio (\(r = \frac{1}{4}\))

Plug these into the formula:\[a_5 = -8 \cdot \left(\frac{1}{4}\right)^4 = -8 \cdot \frac{1}{256} = -\frac{1}{32}\].
Understanding this term specifically demonstrates how sequences shrink or grow over a few steps, emphasizing the significance of the common ratio and the pattern it establishes.