Problem 33 - Algebra and Trigonometry

Question

The first term of a geometric sequence is \(8,\) and the second term is \(4 .\) Find the fifth term.

Step-by-Step Solution

Answer

The fifth term is \( \frac{1}{2} \).

Step 1: Find the Common Ratio

To find the common ratio \( r \) of a geometric sequence, divide the second term by the first term. For the sequence with terms 8 and 4, the common ratio is \( r = \frac{4}{8} = \frac{1}{2} \).

Step 2: Write the General Term Formula

The general term of a geometric sequence can be calculated using the formula \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 3: Substitute Values for the Fifth Term

To find the fifth term \( a_5 \), substitute the known values into the general term formula: \( a_5 = 8 \cdot \left( \frac{1}{2} \right)^{5-1} = 8 \cdot \left( \frac{1}{2} \right)^4 \).

Step 4: Calculate the Fifth Term

Calculate \( \left( \frac{1}{2} \right)^4 = \frac{1}{16} \), then multiply by 8 to get \( a_5 = 8 \cdot \frac{1}{16} = \frac{8}{16} = \frac{1}{2} \). So, the fifth term is \( \frac{1}{2} \).

Key Concepts

Common Ratio

In a geometric sequence, each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. Identifying this ratio is your starting point in solving any problems related to geometric sequences. For example, if your first term is 8 and your second term is 4, you would divide the second term by the first term to find the common ratio: \( r = \frac{4}{8} = \frac{1}{2} \).
This means that each term in the sequence is half of the one before it. Remember, this ratio remains consistent throughout the sequence.

• The common ratio explains how the sequence progresses.
• A positive ratio diminishes, grows, or oscillates the sequence based on its specific value.
• The sequences unfold uniquely depending on decimals or fractions.

Understanding the common ratio is vital because it guides you through calculating other terms in the sequence.

General Term Formula

Once the common ratio is determined, you can calculate any term in a geometric sequence using the general term formula: \( a_n = a_1 \cdot r^{n-1} \). This powerful equation allows you to go from knowing just the initial term and the common ratio to calculating any term number in the sequence.

• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio, which doesn't change.
• \( n \) represents the number of the term you want to find.

By using the formula, any term is within reach. Simply plug in the known values and solve for the term of interest. This formula is particularly useful because it leaps over unknown terms to reach your target directly.

Fifth Term Calculation

The fifth term calculation employs both the common ratio and the general term formula. Having already found that \( r = \frac{1}{2} \), and knowing the first term \( a_1 = 8 \), you can plug these into the formula to find the fifth term \( a_5 \).
Since \( n = 5 \), substitute into the formula: \[ a_5 = 8 \cdot \left( \frac{1}{2} \right)^{5-1} = 8 \cdot \left( \frac{1}{2} \right)^4 \] Now, calculate \( \left( \frac{1}{2} \right)^4 \) as \( \frac{1}{16} \).
Multiply \( \frac{1}{16} \) by 8 to find the fifth term: \[ a_5 = 8 \cdot \frac{1}{16} = \frac{8}{16} = \frac{1}{2} \] Thus, the fifth term is \( \frac{1}{2} \). This stepwise approach simplifies understanding, ensuring clarity and precision in reaching your answer.