Problem 40 - Precalculus: Mathematics for Calculus

Question

Find the indicated term(s) of the geometric sequence with the given description. The first term is \(\frac{1}{12}\) and the second term is \(-\frac{1}{2} .\) Find the sixth term.

Step-by-Step Solution

Answer

The sixth term is -648.

Step 1: Identify the formula

The formula to find any term in a geometric sequence is given by: \[ a_n = a_1 imes r^{(n-1)} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 2: Find the common ratio

Using the information given:\[ a_1 = \frac{1}{12}, \quad a_2 = -\frac{1}{2} \]To find the common ratio \( r \), we use the formula:\[ r = \frac{a_2}{a_1} = \frac{-\frac{1}{2}}{\frac{1}{12}} = -6 \]The common ratio \( r \) is \(-6\).

Step 3: Use the formula to find the sixth term

Now that we know \( a_1 = \frac{1}{12} \) and \( r = -6 \), we can use the formula for the nth term:\[ a_6 = a_1 imes r^{(6-1)} = \frac{1}{12} imes (-6)^5 \]Calculate \((-6)^5\):\[ (-6)^5 = -7776 \]So the sixth term is:\[ a_6 = \frac{1}{12} imes -7776 = -648 \]

Step 4: Conclusion

The sixth term of the geometric sequence is \(-648\).

Key Concepts

Understanding the Common Ratio

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed number called the 'common ratio'.
To find the common ratio, we divide any term by the preceding term.
For the sequence in our exercise, with the first term, \( a_1 = \frac{1}{12} \), and the second term, \( a_2 = -\frac{1}{2} \), the common ratio \( r \) can be calculated as follows:

• Divide the second term by the first term: \( r = \frac{a_2}{a_1} = \frac{-\frac{1}{2}}{\frac{1}{12}} \)


• This simplifies to \( r = -6 \)

So, every term in this geometric sequence is obtained by multiplying the previous term by -6.

Applying the nth Term Formula

The nth term formula is a crucial tool in any geometric sequence. It allows us to find any term in the sequence quickly without having to calculate all the preceding terms.
The formula is:

• \( a_n = a_1 \times r^{(n-1)} \)

In this formula:

• \( a_n \) is the term number we want to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term in the sequence.

This formula simplifies the task of finding any term without recalculating all prior terms, saving time and reducing errors.

Sequence Term Calculation in Practice

To determine the sixth term in the example sequence, we must substitute known values into the nth term formula.
We already know:

• First term, \( a_1 = \frac{1}{12} \)
• Common ratio, \( r = -6 \)
• The term position, \( n = 6 \)

Let's plug these values into the formula: \[ a_6 = \frac{1}{12} \times (-6)^{5} \] Now, calculate \((-6)^5\):

• \((-6) \times (-6) \times (-6) \times (-6) \times (-6) = -7776 \)

Then, calculate the sixth term: \[ a_6 = \frac{-7776}{12} = -648 \] Therefore, the sixth term is \(-648\).
By using these calculations, we navigate through the geometric sequence efficiently.