Problem 34 - Precalculus with Limits

Question

In Exercises 29 - 34, write the first five terms of the geometric sequence. Determine the common ratio and write the \( n \)th term of the sequence as a function of \( n \). \( a_1 = 80, a_{k + 1} = -\dfrac{1}{2} a_k \)

Step-by-Step Solution

Answer

The first five terms of the sequence are 80, -40, 20, -10, 5. The common ratio is -1/2. The nth term of the sequence is \( a_n = 80 * (-\dfrac{1}{2})^{n - 1} \).

Step 1: Generate the first five terms

Using the rule \( a_{k + 1} = -\dfrac{1}{2} a_k \), you can generate the next four terms from the given first term \( a_1 = 80 \) as follows: \n\n\( a_2 = -\dfrac{1}{2} a_1 = -\dfrac{1}{2} * 80 = -40 \)\n\n\( a_3 = -\dfrac{1}{2} a_2 = -\dfrac{1}{2} * -40 = 20 \)\n\n\( a_4 = -\dfrac{1}{2} a_3 = -\dfrac{1}{2} * 20 = -10 \)\n\n\( a_5 = -\dfrac{1}{2} a_4 = -\dfrac{1}{2} * -10 = 5 \)\n\nSo, the first five terms are 80, -40, 20, -10, 5.

Step 2: Determine the common ratio

The common ratio of a geometric sequence can be obtained from any pair of consecutive terms as the quotient \( r = \dfrac{a_{k + 1}}{a_k} \). You can pick the first two terms and apply this formula to get \( r = \dfrac{a_2}{a_1} = \dfrac{-40}{80} = -\dfrac{1}{2} \), which confirms that the common ratio is indeed -1/2.

Step 3: Find the nth term

The nth term of a geometric sequence can be found with the formula \( a_n = a_1 * r^{n - 1} \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of the term. Plugging in the known values, we obtain the nth term as a function of n: \( a_n = 80 * (-\dfrac{1}{2})^{n - 1} \).

Key Concepts

Common Ratio

When working with a geometric sequence, understanding the concept of the **common ratio** is crucial. The common ratio, denoted by \( r \), is a constant factor between consecutive terms of the sequence. To find the common ratio, you simply divide any term by the previous term in the sequence. For example, given the first two terms \( a_1 = 80 \) and \( a_2 = -40 \) from the sequence in the exercise, the common ratio \( r \) is calculated as follows:

• \( r = \frac{a_2}{a_1} = \frac{-40}{80} = -\frac{1}{2} \)

This common ratio of \(-\frac{1}{2}\) tells us that each term is half the magnitude and opposite in sign to the preceding term.
The consistent application of the common ratio lets us generate the entire sequence indefinitely.

Nth Term

Finding the **nth term** of a geometric sequence allows you to determine any term in the sequence without having to list all previous terms. This can be achieved with the formula:\[a_n = a_1 \times r^{n - 1}\]where:

• \( a_n \) is the nth term.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio of the sequence.
• \( n \) is the term number you wish to find.

Starting with the first term, \( a_1 = 80 \), and the common ratio, \( r = -\frac{1}{2} \), you can plug these values into the formula to find any term in the sequence. For instance, to find the 5th term (\( a_5 \)), substitute \( n = 5 \) into the formula:\[a_5 = 80 \times \left(-\frac{1}{2}\right)^{5 - 1} = 80 \times \left(-\frac{1}{2}\right)^4 = 5\]This validates our calculated values earlier, and this formula provides a powerful tool for efficiently determining any term in the sequence.

Terms of a Sequence

The **terms of a sequence** are the individual elements making up the sequence. For a geometric sequence, each term is found by multiplying the previous term by the common ratio. Based on our example sequence:

• \( a_1 = 80 \)
• \( a_2 = -40 \)
• \( a_3 = 20 \)
• \( a_4 = -10 \)
• \( a_5 = 5 \)

These terms clearly illustrate the pattern governed by the common ratio. As a consequence of this pattern:
Each term after the first is sequentially derived and it conforms to the formula \( a_{k+1} = r \cdot a_k \).
This simplicity in determining terms makes geometric sequences easy to analyze and predict, helping us to extend the sequence to as many terms as needed, following the established relationship.