Problem 39 - Precalculus: Mathematics for Calculus

Question

Find the indicated term(s) of the geometric sequence with the given description. The first term is 15 and the second term is \(6 .\) Find the fourth term.

Step-by-Step Solution

Answer

The fourth term is \( \frac{24}{25} \).

Step 1: Identify the common ratio

To find the common ratio ( ), divide the second term by the first term. This will help us identify the factor by which each term is multiplied to get the next term in the sequence. For this sequence: \( r = \frac{6}{15} = \frac{2}{5} \).

Step 2: Use the geometric sequence formula

The general formula for the nth term of a geometric sequence is given by \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. Here, \( a_1 = 15 \) and \( r = \frac{2}{5} \).

Step 3: Calculate the fourth term

Substitute the known values into the formula for the fourth term:\[ a_4 = 15 \cdot \left(\frac{2}{5}\right)^{3}\]First, calculate \( \left(\frac{2}{5}\right)^{3} \):\[ \left(\frac{2}{5}\right)^{3} = \frac{8}{125} \]Then multiply with the first term:\[ a_4 = 15 \cdot \frac{8}{125} = \frac{120}{125} = \frac{24}{25}\]

Step 4: Simplify the result

Finally, simplify \( \frac{24}{25} \) if necessary. In this case, it is already in its simplest form, so \( a_4 = \frac{24}{25} \) is the fourth term of the sequence.

Key Concepts

Understanding the Common Ratio

The common ratio is a key characteristic of any geometric sequence. It describes how each term in the sequence relates to the previous one. To find the common ratio, you simply divide any term in the sequence by the one that precedes it.
In our example, the first term is 15, and the second term is 6. Therefore, to find the common ratio, you would divide 6 by 15:
\[ r = \frac{6}{15} = \frac{2}{5} \]
This fraction, \( \frac{2}{5} \), is the common ratio, meaning each term is \( \frac{2}{5} \) of the one before it.
The common ratio is a critical part of the geometric sequence formula as it determines how the sequence progresses.

• Multiply to find the next term: First term \( \times \frac{2}{5} \).
• Repeat the multiplication to continue the sequence.
• Each term gets continually smaller as shown by the ratio \( \frac{2}{5} \).

Applying the Geometric Sequence Formula

A geometric sequence can be succinctly described using its formula. This formula allows you to find any term in the sequence without listing all prior terms.
The formula is:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Here, \( a_n \) is the specific term you're trying to find, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) represents the position of the term in the sequence.
Let's break it down using our example:

• The first term, \( a_1 = 15 \).
• The common ratio, \( r = \frac{2}{5} \).
• You plug these values into the formula to solve for any \( n \) term you need.

This versatile formula makes it easy to calculate terms further along in the sequence without starting over from the beginning every time.

Performing the nth Term Calculation

Finding the \( n^{th} \) term of a geometric sequence involves using the formula we've learned earlier. Here's how to do it, using the example where we want to find the fourth term.
We have:

• First term \( a_1 = 15 \).
• Common ratio \( r = \frac{2}{5} \).

Now substitute these into the formula:
\[ a_4 = 15 \cdot \left(\frac{2}{5}\right)^{3} \]
Start by calculating the power of the common ratio:
\[ \left(\frac{2}{5}\right)^{3} = \frac{8}{125} \]
Next, multiply this result by the first term:
\[ a_4 = 15 \cdot \frac{8}{125} = \frac{120}{125} \]
Finally, simplify this fraction, if possible. In our case, \( \frac{120}{125} \) simplifies to \( \frac{24}{25} \). This fraction is the fourth term of the sequence.
This method gives you a direct path to finding any term based solely on the first term and the common ratio, avoiding unnecessary calculations.