Problem 37 - 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 of the sequence is \( \frac{1}{2}. \)

Step 1: Identify the Common Ratio

A geometric sequence is defined by a common ratio between consecutive terms. Given that the first term is 8 and the second term is 4, the common ratio \( r \) can be found by dividing the second term by the first term: \[ r = \frac{4}{8} = \frac{1}{2}. \]

Step 2: Use the General Formula for Geometric Sequences

The general 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, \( r \) is the common ratio, and \( n \) is the term number.

Step 3: Substitute Values into the Formula

Substitute the known values into the formula for the fifth term \( (n = 5) \): \[ a_5 = 8 \cdot \left(\frac{1}{2}\right)^{5-1} = 8 \cdot \left(\frac{1}{2}\right)^4. \]

Step 4: Calculate the Power of the Common Ratio

Calculate \( \left(\frac{1}{2}\right)^4 \): \[ \left(\frac{1}{2}\right)^4 = \frac{1}{16}. \]

Step 5: Calculate the Fifth Term

Now, substitute the calculated power value back into the equation for \( a_5 \): \[ a_5 = 8 \cdot \frac{1}{16} = \frac{8}{16} = \frac{1}{2}. \]

Key Concepts

Understanding the Common Ratio

In a geometric sequence, each term after the first is derived by multiplying the previous term by a constant known as the common ratio. The common ratio is crucial in determining the pattern of the sequence and is symbolized by the letter \( r \). For example, if the first term of the sequence is \( 8 \) and the second term is \( 4 \), you can find the common ratio by dividing the second term by the first term:

• \[ r = \frac{4}{8} = \frac{1}{2} \]

This calculation shows that each term is half of the term before it. Recognizing the common ratio provides insight into how the sequence progresses, and it is an essential step in solving problems involving geometric sequences.

Using the nth Term Formula

The nth term formula is a powerful tool for finding any term in a geometric sequence without having to list all the preceding terms. This formula is given by

• \[ a_n = a_1 \cdot r^{n-1} \]

where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number you want to find. By substituting the known values into this formula, such as \( a_1 = 8 \) and \( r = \frac{1}{2} \), you can compute the 5th term (or any other term):

• \[ a_5 = 8 \cdot \left(\frac{1}{2}\right)^{5-1} \]

This allows you to efficiently solve your problem without unnecessary calculations.

Geometric Series Calculation

Although this exercise primarily dealt with finding a specific term in a sequence, geometric series calculations also play a key role in understanding how sums of terms operate within a geometric framework. A geometric series is the sum of the terms of a geometric sequence. When you know the first term and the common ratio, you can determine the sum of the first \( n \) terms using:

• \[ S_n = a_1 \cdot \frac{1-r^n}{1-r} \]

where \( S_n \) denotes the sum of the first \( n \) terms. This formula is very useful when you need to find the total of multiple terms quickly, for instance, the sum of the first 5 terms in our sequence. Always remember to validate that \( r eq 1 \) as this formula does not hold otherwise. Understanding this concept helps you go beyond individual terms to grasp the overarching patterns of geometric sequences.