Problem 16 - 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. $$a_{1}=-6, r=\frac{2}{3}$$

Step-by-Step Solution

Answer

Answer: The sixth term of the geometric sequence is -192/243, and the general formula for the nth term is \(a_n = -6 (\frac{2}{3})^{(n-1)}\).

Step 1: Identify the given values

The first term \(a_1\) is given as \(-6\), and the common ratio \(r\) is given as \(\frac{2}{3}\).

Step 2: Find the sixth term using the formula

To find the sixth term, \(a_6\), we need to use the formula \(a_n = a_1 \cdot r^{(n-1)}\) and substitute \(n=6\), \(a_1 = -6\), and \(r = \frac{2}{3}\): \(a_6 = -6 (\frac{2}{3})^{(6-1)} = -6 (\frac{2}{3})^{5}\) Now, calculate the value of \(a_6\): \(a_6 = -6 (\frac{2^5}{3^5}) = -6 (\frac{32}{243}) = -\frac{192}{243}\) So, the sixth term, \(a_6\), is \(-\frac{192}{243}\).

Step 3: Find the formula for the nth term

Now we can write the formula for the nth term, \(a_n\), using the given values of \(a_1\) and \(r\): \(a_n = -6 (\frac{2}{3})^{(n-1)}\) So, the general formula for the nth term of the geometric sequence is \(a_n = -6 (\frac{2}{3})^{(n-1)}\).

Key Concepts

nth term formula

In the realm of geometric sequences, finding the nth term is a fundamental skill. A geometric sequence is an ordered list of numbers in which each term after the first is found by multiplying the previous term by a certain number, called the common ratio. To express this relationship in mathematical terms, we use the nth term formula:

• The formula is: \(a_n = a_1 \cdot r^{(n-1)}\)
• Here, \(a_n\) represents the nth term you're trying to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number of the sequence you're looking for.

This formula allows you to calculate any term in the sequence without having to list all the preceding terms, a tool particularly handy in long sequences.

common ratio

The common ratio in a geometric sequence is a pivotal concept. It is the fixed number that you multiply each term by to get to the next term. It defines the pattern and progression of the sequence. Here's how it works:

• The common ratio, denoted as \(r\), remains constant throughout the sequence.
• If \(r > 1\), each term will be larger than the previous one; the sequence is increasing.
• If \(0 < r < 1\), each term will be smaller than the previous one; the sequence is decreasing.
• If \(r = 1\), all terms in the sequence are the same.
• If \(r < 0\), terms will alternate between positive and negative.

Understanding the common ratio helps predict how the sequence will progress and calculate future terms efficiently.

sequence term calculation

To calculate a specific term in a geometric sequence, you'll use the nth term formula, as we've explored. Let's break it down further with an example calculation:

• You start with the formula: \(a_n = a_1 \cdot r^{(n-1)}\)
• Suppose you know that \(a_1 = -6\) and the common ratio \(r = \frac{2}{3}\).
• To find the 6th term (as requested in an exercise), substitute \(n = 6\) into the formula.
• That gives you: \(a_6 = -6 \cdot \left(\frac{2}{3}\right)^{5}\)
• Calculate \(\left(\frac{2}{3}\right)^5\), which is \(\frac{32}{243}\).
• Then multiply: \(-6 \times \frac{32}{243} = -\frac{192}{243}\).

So, using the sequence term calculation, you arrive at \(a_6 = -\frac{192}{243}\). This approach removes guesswork and provides a systematic way to determine any term in the sequence.