Problem 13 - Algebra for College Students

Question

\text { The } 9 \text { th term of } 8,4,2,1, \ldots, \frac{1}{32}

Step-by-Step Solution

Answer

The 9th term is \( \frac{1}{32} \).

Step 1: Identify the Sequence Type

The given sequence is 8, 4, 2, 1, ..., \( \frac{1}{32} \). By observing the constant ratio between consecutive terms, we recognize that this is a geometric sequence.

Step 2: Determine the Common Ratio

To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{4}{8} = \frac{1}{2} \). Therefore, the common ratio \( r \) is \( \frac{1}{2} \).

Step 3: Use the Formula for the nth Term of a Geometric Sequence

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

Step 4: Calculate the 9th Term

The first term \( a_1 \) is 8. Plug in \( n = 9 \), \( a_1 = 8 \), and \( r = \frac{1}{2} \) into the formula: \( a_9 = 8 \times \left( \frac{1}{2} \right)^{9-1} = 8 \times \left( \frac{1}{2} \right)^8 = 8 \times \frac{1}{256} = \frac{1}{32} \).

Key Concepts

Common Ratio

In the context of a geometric sequence, the **common ratio** is a crucial element. This is the factor by which you multiply a term in the sequence to get the next term. To find it, you simply divide any term by the previous term.

In our exercise, we have the sequence: 8, 4, 2, 1, ..., \( \frac{1}{32} \). To find the common ratio \( r \), take the second term (4) and divide it by the first term (8). This gives us,

• \( r = \frac{4}{8} = \frac{1}{2} \)

So, each term is half of the previous one. Understanding the common ratio is key as it allows us to determine the type of sequence and predict future terms. In this sequence, the common ratio is constant, confirming the sequence is geometric.

Nth Term Formula

The nth term formula in a geometric sequence helps us find any term in the sequence without listing all previous terms. The formula is:

• \( a_n = a_1 \times r^{n-1} \)

Where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.

In the example provided, the first term \( a_1 \) is 8, and the common ratio \( r \) is \( \frac{1}{2} \). We can use the formula to find any term in the sequence. For the 9th term \( a_9 \), plug in these values:

• \( a_9 = 8 \times \left( \frac{1}{2} \right)^{9-1} = 8 \times \left( \frac{1}{2} \right)^8 = \frac{1}{32} \)

This formula is powerful because it allows calculation of any term directly from its position.

Sequence Type Identification

**Identifying the sequence type** is an essential first step in solving problems involving series of numbers. A geometric sequence is characterized by a constant ratio between consecutive terms. Examining the given sequence, 8, 4, 2, 1, ..., reveals a repeating division by 2. This signifies that the sequence is geometric.

Recognizing the sequence type is foundational because it informs which formulas and strategies will be most efficient for solving related problems. In a geometric sequence, using methods like the common ratio and nth term formula becomes directly applicable. Thus, these two observations not only help in verifying the sequence's nature but also in problem-solving.