Problem 53 - College Algebra

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 First Term

In a geometric sequence, the first term is denoted by \(a\). From the given sequence \(\{b, 4b, 16b, \ldots\}\), the first term \(a\) is \(b\).

Step 2: Find the Common Ratio

The common ratio \(r\) is found by dividing the second term by the first term. In this sequence, \(4b \div b = 4\). Therefore, the common ratio \(r\) is 4.

Step 3: Use the Geometric Sequence Formula

The \(n^{\text{th}}\) term of a geometric sequence can be found using the formula \(a_n = a \cdot r^{n-1}\).

Step 4: Substitute Values into the Formula

To find the 5th term, substitute \(a = b\), \(r = 4\), and \(n = 5\) into the formula: \(a_5 = b \cdot 4^{5-1}\).

Step 5: Calculate the 5th Term

Calculate \(4^{4}\): \(4 \times 4 \times 4 \times 4 = 256\). Therefore, the \(5^{\text{th}}\) term is \(b \cdot 256 = 256b\).

Key Concepts

First Term

In a geometric sequence, the first term is of utmost importance, as it serves as the starting point for the entire sequence. This term is usually denoted by the letter \(a\). The given sequence, \(\{b, 4b, 16b, \ldots\}\), clearly shows that the first term \(a\) is simply \(b\).

To identify the first term, observe the sequence and see what is given in the very first position. Knowing the first term helps in finding other terms in the sequence, as it forms the base of our calculations.

Remember, without a clear understanding of the first term, it will be difficult to determine the rest of the terms in the sequence.

Common Ratio

The common ratio in a geometric sequence is a vital element. It determines the factor by which each term of the sequence is multiplied to get the next term. To find this ratio, divide the second term by the first term.

For the sequence \(\{b, 4b, 16b, \ldots\}\), the second term is \(4b\). Dividing \(4b\) by the first term \(b\), gives us \(4b/b = 4\). Thus, the common ratio \(r\) is 4.

The common ratio must remain constant throughout the sequence. This consistent multiplication is what helps define the sequence as geometric. Having a grasp of the common ratio makes it easier to use the geometric sequence formula effectively.

Geometric Sequence Formula

The geometric sequence formula is an essential component for calculating any term in the sequence. It is given as:\(a_n = a \cdot r^{n-1}\).

Here's a breakdown of what each symbol represents:

• \(a_n\): The \(n^{th}\) term of the sequence you wish to find.
• \(a\): The first term of the sequence.
• \(r\): The common ratio.
• \(n\): The term number you're interested in calculating, denoted in the sequence.

This formula allows you to calculate any term in a geometric sequence quickly and precisely. Just plug in the first term, common ratio, and the desired term number, and the formula will do the rest of the work for you.

N-th Term Calculation

Calculating the \(n^{th}\) term in a geometric sequence brings together all the previous elements like the first term, common ratio, and the geometric sequence formula. Here's how it works in practice.

We need to find the \(5^{th}\) term in the sequence \(\{b, 4b, 16b, \ldots\}\). With \(a = b\), \(r = 4\), and \(n = 5\), the formula \(a_n = a \cdot r^{n-1}\) becomes \(a_5 = b \cdot 4^{5-1}\).

First, calculate \(4^{4}\) as \(4 \times 4 \times 4 \times 4 = 256\). Then multiply this by the first term \(b\), giving \(a_5 = b \cdot 256 = 256b\).

The \(5^{th}\) term is therefore \(256b\). Understanding this process of finding the \(n^{th}\) term will allow you to explore and solve for any term in a similar sequence easily.