Problem 41 - Precalculus Mathematics for Calculus

Question

Which term of the geometric sequence \(2,6,18, \ldots\) is \(118,098 ?\)

Step-by-Step Solution

Answer

The term is the 11th term in the sequence.

Step 1: Identify Given Information

The given geometric sequence is \(2, 6, 18, \ldots\). You need to find which term in this sequence equals \(118,098\).

Step 2: Find the Common Ratio

In a geometric sequence, the common ratio \(r\) can be found by dividing the second term by the first term. Here, \(r = \frac{6}{2} = 3\).

Step 3: Write the nth-term Formula

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 and \(r\) is the common ratio.

Step 4: Set Up the Equation

You know \(a_n = 118,098\), \(a_1 = 2\), and \(r = 3\). Substitute these into the nth-term formula: \(118,098 = 2 \cdot 3^{n-1}\).

Step 5: Solve for n

First, divide both sides by 2: \(59,049 = 3^{n-1}\). Now, solve for \(n-1\) by finding what power of 3 gives \(59,049\). This can be done through trial and error or using logarithms: \(n-1 = \log_3{59,049}\). It turns out that \(59,049 = 3^{10}\), so \(n-1=10\).

Step 6: Find n

Add 1 to \(n-1\) to solve for \(n\): \(n = 11\).

Key Concepts

Understanding the Common Ratio in a Geometric Sequence

The common ratio is an essential aspect of any geometric sequence. It's what makes a geometric series tick. This ratio, usually denoted as \( r \), is the factor by which we multiply one term to get the next term in the sequence.
To find the common ratio in a geometric sequence, simply divide any term by the previous term. For instance, with the sequence \( 2, 6, 18, \ldots \), we can find \( r \) by dividing the second term by the first term:

• \( r = \frac{6}{2} = 3 \)

This means each term is three times the previous term. Knowing the common ratio is crucial as it helps us understand the structure and growth of the sequence. It also plays a major role in determining any term in the sequence.

Navigating the nth-term Formula for Geometric Sequences

To find a specific term in a geometric sequence, we use the nth-term formula. This formula enables us to calculate any term without listing all the previous terms.
The nth-term formula is:

• \( a_n = a_1 \cdot r^{n-1} \)

Here,

• \( a_n \) represents the term we want to find.
• \( a_1 \) is the first term in the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term in the sequence.

Understanding this formula is like unlocking a map to the sequence. We simply plug in the values we know - for example, in our sequence \( 2, 6, 18, \ldots \), to find \( a_n \), knowing \( a_1 = 2 \) and \( r = 3 \), we just substitute and solve. This formula is a shortcut that saves time and effort.

Solving for the nth Term in a Geometric Sequence

Solving for the nth term can feel like detective work, where we put together clues to find the unknown. It requires using both the common ratio and the nth-term formula.
In the problem, we're searching for which term equals 118,098 in the sequence \(2, 6, 18, \ldots\). Begin by setting up the nth-term equation:

• \( 118,098 = 2 \cdot 3^{n-1} \)

This equation arises from substituting the known values into the nth-term formula. Next,

• Divide both sides by 2 to isolate the exponential part: \( 59,049 = 3^{n-1} \).
• Solve for \( n-1 \) by finding what power of 3 equals 59,049. You can do this using trial and error or logarithms.
• Find that \( 59,049 = 3^{10} \), meaning \( n-1 = 10 \).
• Finally, add 1 to \( n-1 \) to determine \( n = 11 \).

So, the 11th term in the sequence is 118,098. Solving for \(n\) empowers you to leap right to any term in the sequence.