Problem 18 - Contemporary Precalculus

Question

In Exercises \(13-22,\) one term and the common ratio r of a geometric sequence are given. Find the sixth term and a formula for the nth term. $$a_{1}=\pi, r=\frac{1}{5}$$

Step-by-Step Solution

Answer

Answer: The sixth term of the geometric sequence is \(\pi \cdot \left(\dfrac{1}{5}\right)^{5}\) and the formula for the nth term is given by \(a_{n} = \pi \cdot \left(\dfrac{1}{5}\right)^{n-1}\).

Step 1: Find the sixth term of the sequence

To find the sixth term, we will use the general formula for the nth term of a geometric sequence and plug in the values of \(a_{1}\), \(r\), and \(n=6\). $$ a_{6} = a_{1} \cdot r^{6-1} = \pi \cdot \left(\dfrac{1}{5}\right)^{5} $$

Step 2: Write a formula for the nth term of the sequence

We already have the general formula for the nth term of a geometric sequence. We will simply plug in the given values of \(a_{1}\) and \(r\) as follows: $$ a_{n} = a_{1} \cdot r^{n-1} = \pi \cdot \left(\dfrac{1}{5}\right)^{n-1} $$

Step 3: Answers

The sixth term of the geometric sequence is \(\pi \cdot \left(\dfrac{1}{5}\right)^{5}\) and the formula for the nth term is given by \(a_{n} = \pi \cdot \left(\dfrac{1}{5}\right)^{n-1}\).

Key Concepts

nth term formula

The \(n\)th term formula of a geometric sequence is incredibly handy. It allows you to find any term in the sequence without listing all preceding terms. The formula is defined as: \[a_n = a_1 imes r^{(n-1)}\]Here, \(a_n\) represents the term you are trying to find in the sequence. \(a_1\) is the first term, which is given, and \(r\) is the common ratio, also provided. The \(n-1\) part represents one less than the term number you are calculating.

• \(a_n\): The \(n\)th term you wish to find.
• \(a_1\): The first term of the sequence.
• \(r\): The common ratio that each term is multiplied by to get the next term.

To use this formula effectively, you simply plug the given values into it. For example, if you need the sixth term and know that \(a_1\) is \(\pi\) and \(r\) is \(\frac{1}{5}\), you substitute these into the formula and calculate accordingly.

common ratio

The common ratio is a vital concept in understanding geometric sequences. It defines how the sequence progresses. In a geometric sequence, each term is derived by multiplying the previous term by the common ratio \(r\). This consistent multiplication results in the formation of a geometric sequence. When given a common ratio of \(\frac{1}{5}\), it indicates that each term is \(1/5\) of the previous term. This ratio can be any real number, positive or negative, and can also be a fraction. A correct understanding of the common ratio helps you to predict the nature of the sequence and the value of different terms. - Small values \(r < 1\) result in a diminishing sequence.- Values \(r = 1\) lead to a constant sequence.- Greater values \(r > 1\) produce a growing sequence.By identifying the common ratio, you gain insights into the behavior of the sequence and can apply this to find specific terms.

sequence term calculation

Calculating terms in a geometric sequence becomes straightforward by using the \(n\)th term formula. To find any particular term, such as the sixth, you can apply the formula: \[a_n = a_1 \times r^{(n-1)}\]Take the given values, where \(a_1 = \pi\) and the common ratio \(r = \frac{1}{5}\), and substitute these into the formula to compute \(a_6\): \[a_6 = \pi \times \left(\frac{1}{5}\right)^{(6-1)} = \pi \times \left(\frac{1}{5}\right)^5\]Break down this process:

• Identify the first term \(a_1\) and the common ratio \(r\).
• Plug these into the formula alongside the term number minus one, which is \(6 - 1 = 5\) for the sixth term.
• Multiply the initial term by the common ratio raised to the power calculated.

By following these steps, the calculated term provides the specific sequence value that can contribute to more complex problem-solving contexts.