Problem 28 - Precalculus: Mathematical for Calculus

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

Common ratio is \( \frac{1}{4} \); fifth term is \(-\frac{1}{32} \); nth term is \( a_n = -8 \left(\frac{1}{4}\right)^{(n-1)} \).

Step 1: Determine the Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio \( r \). To find \( r \), divide the second term by the first term: \[ r = \frac{-2}{-8} = \frac{1}{4}. \] Verify this ratio by checking other consecutive terms: \[ r = \frac{-\frac{1}{2}}{-2} = \frac{1}{4}. \] Thus, the common ratio is \( \frac{1}{4}. \)

Step 2: Calculate the Fifth Term of the Sequence

To find the fifth term of a geometric sequence, use the formula \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio. Here, \( n = 5 \). Therefore, \[ a_5 = -8 \cdot \left(\frac{1}{4}\right)^{5-1} = -8 \cdot \left(\frac{1}{4}\right)^4 = -8 \cdot \frac{1}{256} = -\frac{1}{32}. \] The fifth term is \(-\frac{1}{32}\).

Step 3: Express the General Formula for the nth Term

The general formula for the \( n \)th term of a geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \). Substituting the known values, the sequence formula becomes: \[ a_n = -8 \cdot \left(\frac{1}{4}\right)^{(n-1)}. \] This formula can be used to find any term in the sequence.

Key Concepts

Common Ratio

In a geometric sequence, understanding the **common ratio** is key because it tells us how each term in the sequence relates to the previous one. Think of the common ratio as the factor you multiply a term by to get the next term in the sequence. For example, if we look at our sequence: -8, -2, -\(\frac{1}{2}\), -\(\frac{1}{8}\), ...We find the common ratio by dividing a term by the one before it. In this case, \[ r = \frac{-2}{-8} = \frac{1}{4}. \]To check our work, let's verify the ratio with other terms:- From \(-2\) to \(-\frac{1}{2}\): \[ r = \frac{-\frac{1}{2}}{-2} = \frac{1}{4}. \]When your ratios are consistent, you've found your common ratio! In this sequence, it's \(\frac{1}{4}\). This consistency confirms that you're following the pattern correctly.

nth Term Formula

Once you know the common ratio, you can find any term in the sequence using the **nth term formula**. The formula is \[ a_n = a_1 \cdot r^{(n-1)} \]where:

• \(a_n\) is the term you want to find,
• \(a_1\) is the first term,
• \(r\) is the common ratio,
• \(n\) is the term number.

Let's apply this formula to find the fifth term (\(a_5\)) of our sequence:\[ a_5 = -8 \cdot \left(\frac{1}{4}\right)^4 = -8 \cdot \frac{1}{256} = -\frac{1}{32}. \]This method allows you to quickly find not just the fifth term but any term in the sequence. By just plugging in the term number \(n\), you can compute its value with ease.

Sequence Analysis

**Sequence analysis** involves looking closely at a pattern to understand how it develops. In geometric sequences, the relationship between terms is rooted in multiplication. By analyzing a sequence, you can predict future terms, identify patterns, and even derive the formula. For our sequence, observing the values helps us see:

• Each term is negative, following the negative nature of the first term.
• The terms decrease in absolute value, showing division by the common ratio \(\frac{1}{4}\).
• The sequence representation with the nth term formula: \[ a_n = -8 \cdot \left(\frac{1}{4}\right)^{(n-1)} \]
By understanding these patterns, we can make informed guesses, computations, and derive meaningful conclusions about the data's behavior in this and similar sequences.