Problem 53 - College Algebra with Corequisite Support

Question

Find the \(5^{\text {th }}\) term of the geometric sequence $$ \\{b, 4 b, 16 b, \ldots\\} $$

Step-by-Step Solution

Answer

The 5th term is \(256b\).

Step 1: Identify the Sequence

The given sequence is \(b, 4b, 16b, \ldots\). This is a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio.

Step 2: Find the Common Ratio

To find the common ratio, \(r\), divide the second term by the first term: \(r = \frac{4b}{b} = 4\). This means each term is multiplied by 4 to get the next term.

Step 3: Use the Formula for the n-th Term

The formula for the n-th term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 4: Calculate the 5th Term

Substitute the known values into the formula: \(a_5 = b \cdot 4^{5-1} = b \cdot 4^4\).

Step 5: Compute the Power of the Common Ratio

Calculate \(4^4 = 256\). This will be multiplied by the first term.

Step 6: Finalize the 5th Term

Now, substitute back to get \(a_5 = b \cdot 256 = 256b\). Thus, the 5th term of the sequence is \(256b\).

Key Concepts

Understanding the Common Ratio in Geometric Sequences

In a geometric sequence, the common ratio is the constant factor that you multiply by to get from one term to the next. It's a key element that defines the structure of a geometric sequence and distinguishes it from other types of sequences.
For example, if you have a sequence like \(b, 4b, 16b, \ldots\), one way to find the common ratio \(r\) is by dividing the second term by the first term. This will give you the ratio between consecutive terms. In our example, the common ratio is \(r = \frac{4b}{b} = 4\).
This means that every term is 4 times the previous one. Recognizing the common ratio allows you to predict subsequent terms in the sequence and use formulas effectively. Understanding this concept will make calculating any term in a geometric sequence straightforward.

Using the nth Term Formula for Geometric Sequences

The nth term formula is crucial for finding any term in a geometric sequence without listing them all out. This formula is written as:
\[a_n = a_1 \cdot r^{n-1}\]where \(a_n\) represents the nth term that you want to find,

• \(a_1\) is the first term in the sequence.
• \(r\) is the common ratio you found earlier.
• \(n\) is the position in the sequence of the term you are looking for.

In our earlier example, if you were finding the 5th term, you'd use this formula to solve it:
Substitute \(b\) for \(a_1\) and 4 for \(r\). This gives \(a_5 = b \cdot 4^{5-1} = b \cdot 4^4\). Understanding this formula allows you to efficiently calculate any term in a sequence, no matter how large \(n\) is.

Sequence Calculation: Finding Specific Terms

Calculating specific terms in a sequence is a matter of substituting known values into the nth term formula. For effective calculations, follow these steps:

• Identify the first term \(a_1\) and the common ratio \(r\).
• Determine the term position \(n\) you want to find in the sequence.
• Substitute these values into the nth term formula \(a_n = a_1 \cdot r^{n-1}\).
• Simplify by calculating the power \(r^{n-1}\).
• Multiply this result by \(a_1\) to find your desired term \(a_n\).

Taking our example: we calculated the 5th term \(a_5\) by identifying that \(a_1 = b\), \(r = 4\), and \(n = 5\). Thus, \(a_5 = b \cdot 4^4\ = 256b\).
By following these steps, you'll quickly and accurately find specific terms in any geometric sequence, ensuring clarity and confidence in your calculations.