Problem 37 - Precalculus Mathematics for Calculus

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: Understanding the Problem

We have a geometric sequence where the first term \(a_1 = 8\) and the second term \(a_2 = 4\). Our goal is to find the fifth term \(a_5\) of this sequence.

Step 2: Identifying the Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed number called the common ratio \(r\). We can find \(r\) using the first and second terms:\[r = \frac{a_2}{a_1} = \frac{4}{8} = \frac{1}{2}.\]

Step 3: Writing the General Formula for Geometric Sequence

The general formula for the \(n\)-th term \(a_n\) of a geometric sequence is:\[a_n = a_1 \cdot r^{n-1}.\] Here, \(a_1 = 8\) and \(r = \frac{1}{2}.\)

Step 4: Calculating the Fifth Term

Substitute the values \(a_1 = 8\), \(r = \frac{1}{2}\), and \(n = 5\) into the formula:\[a_5 = 8 \cdot \left(\frac{1}{2}\right)^{5-1} = 8 \cdot \left(\frac{1}{2}\right)^4.\]

Step 5: Simplifying the Expression

To calculate \(\left(\frac{1}{2}\right)^4\), raise the fraction to the fourth power:\[\left(\frac{1}{2}\right)^4 = \frac{1}{16}.\]Now substitute back:\[a_5 = 8 \times \frac{1}{16} = \frac{8}{16} = \frac{1}{2}.\]

Step 6: Conclusion

The fifth term \(a_5\) of the geometric sequence is \(\frac{1}{2}\).

Key Concepts

Common Ratio

In a geometric sequence, every consecutive term is generated by multiplying the previous term by a constant, known as the *common ratio*. This common ratio, represented by \( r \), remains the same throughout the sequence.
For example, to determine the common ratio of a sequence where the first term \( a_1 = 8 \) and the second term \( a_2 = 4 \), you need to divide the second term by the first term:

• \( r = \frac{a_2}{a_1} = \frac{4}{8} = \frac{1}{2} \)

Once you know the common ratio, you can generate any term in the sequence by continuing to multiply by \( r \).
Understanding and identifying the common ratio is crucial as it enables you to apply the sequence's general formula to find other terms.

Sequence General Formula

To find any term in a geometric sequence, the sequence general formula comes in handy. The general formula for the \( n \)-th term of a geometric sequence allows you to calculate the value of any term if you know the first term and the common ratio.
The formula is:

• \( a_n = a_1 \cdot r^{n-1} \),

where:

• \( a_n \) is the \( n \)-th term,
• \( a_1 \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the term number.

For instance, to calculate the fifth term in our sequence where \( a_1 = 8 \) and \( r = \frac{1}{2} \), plug in the values:

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

This formula provides a powerful tool for predicting the behavior of the sequence over time.

Exponentiation

Exponentiation is a fundamental part of the sequence formula within geometric sequences. It involves raising a number, known as the base, to the power of an exponent. The exponent indicates how many times the base will be multiplied by itself.
In the formula \( a_n = a_1 \cdot r^{n-1} \), you use exponentiation to determine the progression of the sequence. If we want to find the fifth term in our example, we calculate \((\frac{1}{2})^4\):

• \( \left(\frac{1}{2}\right)^4 = \frac{1}{16} \)

Exponentiation can significantly change the outcome, especially as the power increases. Therefore, grasping how exponentiation impacts a series is key to successfully applying the geometric sequence formula.
This understanding allows you to predict and confidently calculate the terms of the sequence.