Problem 30 - Precalculus: Mathematics for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$7, \frac{14}{3}, \frac{28}{9}, \frac{56}{27}, \dots$$

Step-by-Step Solution

Answer

Common ratio = \( \frac{2}{3} \); Fifth term = \( \frac{112}{81} \); \( n \)th term = \( 7(\frac{2}{3})^{n-1} \).

Step 1: Identify the First Term

In a geometric sequence, the first term is denoted by \( a \). For the given sequence, the first term \( a = 7 \).

Step 2: Calculate the Common Ratio

The common ratio \( r \) in a geometric sequence is the factor that each term is multiplied by to get the next term. To find \( r \), divide the second term by the first term: \( r = \frac{ \frac{14}{3} }{ 7 } \). Simplifying gives \( r = \frac{14}{3} \times \frac{1}{7} = \frac{2}{3} \).

Step 3: Find the Fifth Term

To find the fifth term, use the formula for the \( n \)th term of a geometric sequence: \( a_n = ar^{(n-1)} \). For \( n = 5 \), \( a_5 = 7 \times \left( \frac{2}{3} \right)^{4} \). Calculate \( \left( \frac{2}{3} \right)^{4} = \frac{16}{81} \). So, \( a_5 = 7 \times \frac{16}{81} = \frac{112}{81} \).

Step 4: Derive the General Formula for the \( n \)th Term

The formula for the \( n \)th term of a geometric sequence is \( a_n = ar^{(n-1)} \). With \( a = 7 \) and \( r = \frac{2}{3} \), the formula becomes \( a_n = 7\left(\frac{2}{3}\right)^{n-1} \).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a crucial element. It tells us what number we multiply each term by to get the next one. In our original problem, we see the sequence starts with 7 and continues with \( \frac{14}{3} \), \( \frac{28}{9} \), and \( \frac{56}{27} \). To find the common ratio \( r \), you need to divide any term by its preceding term.

The original solution calculated it as follows:

• Take the second term \( \frac{14}{3} \) and divide it by the first term 7: \( r = \frac{ \frac{14}{3} }{7} \).
• Simplify the fraction by multiplying by the reciprocal of 7, giving: \( r = \frac{14}{3} \times \frac{1}{7} = \frac{2}{3} \).

Each term in the sequence is \( \frac{2}{3} \) times the term before it. The common ratio not only defines the sequence's pattern but also allows us to calculate any term's value when combined with the first term.

Exploring the nth Term Formula

The nth term formula in a geometric sequence is a tool that provides the term at any position in the sequence. It's expressed generally as \( a_n = ar^{(n-1)} \). Let's break down what this means.

• \( a_n \) represents the nth term you are looking to find.
• \( a \) is the first term of the sequence, which in this case is 7.
• \( r \) is the common ratio - for our sequence, we found it to be \( \frac{2}{3} \).
• \( (n-1) \) is the position of the term minus one, since the sequence starts at the power of zero.

With this formula, you can calculate any term in the sequence simply by plugging in the values for \( a \), \( r \), and \( n \). For example, if you wanted to find the 10th term, substitute in the formula: \( a_{10} = 7(\frac{2}{3})^{9} \).

This formula is essential for exploring and predicting future terms in any geometric sequence.

Calculating the Fifth Term

Once you understand the nth term formula, determining a specific term like the fifth term becomes straightforward. For our sequence, we've already used the formula to calculate the fifth term. According to the step-by-step solution, you start with \( n = 5 \), and plug the values into the formula \( a_n = ar^{(n-1)} \).

Here's how it's done:

• The original formula for the nth term is \( a_n = 7\left(\frac{2}{3}\right)^{n-1} \).
• Substitute 5 for n: \( a_5 = 7\left(\frac{2}{3}\right)^{4} \).
• Calculate \( \left(\frac{2}{3}\right)^{4} = \frac{16}{81} \).
• Finally, multiply by 7: \( a_5 = 7 \times \frac{16}{81} = \frac{112}{81} \).

Now you have \( a_5 = \frac{112}{81} \), the fifth term in the sequence. Understanding this specific calculation helps reinforce your understanding of how to navigate through any term in a geometric sequence.