Problem 28 - Contemporary Precalculus

Question

In Exercises \(23-30,\) show that the given sequence is geometric and find the common ratio. $$\left\\{4^{n-4}\right\\}$$

Step-by-Step Solution

Answer

Answer: Yes, the sequence {4^(n-4)} is geometric with a common ratio of 4.

Step 1: Identify the sequence and its general term

The given sequence is \(\left\\{4^{n-4}\right\\}\). The general term \(a_n\) can be written as: $$a_n = 4^{n-4}$$

Step 2: Calculate the ratio between consecutive terms

To determine if the sequence is geometric, we need to find the ratio between consecutive terms, \(r = \frac{a_{n+1}}{a_n}\). Using the general term formula, we can find \(a_{n+1}\): $$a_{n+1} = 4^{(n+1)-4} = 4^{n-3}$$ Now, we can find the ratio \(r\): $$r = \frac{4^{n-3}}{4^{n-4}}$$ To simplify this ratio, we can use the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\): $$r = 4^{(n-3)-(n-4)} = 4^{1}$$

Step 3: Determine if the sequence is geometric and find the common ratio

Since the ratio between consecutive terms is constant (\(r = 4\)), the sequence is geometric. Therefore, the common ratio for the sequence \(\left\\{4^{n-4}\right\\}\) is \(4\).

Key Concepts

Common Ratio

Understanding the common ratio is key to analyzing geometric sequences. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant, known as the common ratio. For the sequence \( \{4^{n-4}\} \), the common ratio is discovered by dividing one term by the previous term.
The formula for determining the common ratio \( r \) is given by:

• \( r = \frac{a_{n+1}}{a_n} \)

Applying this formula to our sequence, we found \( r = 4 \). This constant ratio of 4 indicates that each term is four times the previous term, confirming the sequence's geometric nature.

Exponent Rules

Simplifying terms in geometric sequences often requires a good grasp of exponent rules. In our exercise, the exponent rule used is:

• \( \frac{a^m}{a^n} = a^{m-n} \)

This rule allows us to easily handle powers when dividing exponential expressions. For example, with the sequence term \( \frac{4^{n-3}}{4^{n-4}} \), subtracting the exponents gives \( 4^{1} \), simplifying the ratio to 4.
Exponent rules like this make it easier to work with sequences and identify patterns quickly.

Sequences

Sequences are ordered lists of numbers following a particular rule. A geometric sequence is a type of sequence where the ratio between consecutive terms remains the same.
To verify if a sequence is geometric, note whether this ratio is constant across the sequence. In this case, the sequence \( \{4^{n-4}\} \) shows a constant ratio of 4.

• Each term is derived by multiplying the previous term by this ratio.
• Understanding the type and behavior of sequences aids in prediction and calculation of future terms.

Recognizing patterns in sequences is crucial for solving problems and interpreting data efficiently.