Problem 37 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The 11th term of the sequence is 118,098.

Step 1: Identify the common ratio

A geometric sequence has a constant ratio between consecutive terms. To find the ratio, divide the second term by the first term. Therefore, the common ratio \( r \) is \( r = \frac{6}{2} = 3 \).

Step 2: Use the formula for the nth term of a geometric sequence

The formula for the nth term in a geometric sequence is given by \( a_n = a_1 \, r^{n-1} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 3: Set up the equation

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

Step 4: Solve for \( n \)

First, divide both sides by 2 to isolate the power: \[3^{n-1} = \frac{118,098}{2} = 59,049.\]Then, identify that \( 59,049 = 3^{10} \). Thus, we have \( 3^{n-1} = 3^{10} \).

Step 5: Find the value of \( n \)

Since the bases are the same, set \( n-1 = 10 \). Solving for \( n \) gives:\[n = 10 + 1 = 11.\]

Key Concepts

Common Ratio

The common ratio in a geometric sequence is a fundamental concept to grasp. It is the factor by which each term is multiplied to obtain the next term in the sequence.
Identifying the common ratio helps in understanding the growth pattern of the sequence. To find the common ratio, simply divide a term by its preceding term.

For example, in the sequence given in the exercise, the terms are 2, 6, and 18. Thus, we find the common ratio by dividing the second term by the first term:

• \(r = \frac{6}{2} = 3\)
• Verifying with the next consecutive terms: \(\frac{18}{6} = 3\)

This confirms the pattern and establishes that every term is multiplied by 3 to obtain the next one.

nth Term of a Geometric Sequence

The nth term of a geometric sequence is an important tool used to find any specific term in the sequence, especially when the terms grow or shrink rapidly. This is expressed using a formula which is crucial for solving such problems:

\[ a_n = a_1 \, r^{n-1} \] Here:

• \(a_n\) is the nth term we want to find
• \(a_1\) is the first term of the sequence
• \(r\) is the common ratio
• \(n\) is the specific term number

Using this formula, you can pinpoint any term in the sequence by plugging in the values of the known parameters.
For the example given, the first term \(a_1\) is 2, the common ratio \(r\) is 3, and we need to find the term when \(a_n = 118,098\). This forms the equation to solve.

Solving Equations

Solving equations involving geometric sequences often requires manipulating the known formula to find the unknown term number \(n\). This involves transforming the formula based on given values and performing algebraic operations:

• You start with \(a_n = a_1 \, r^{n-1}\).
• Substitute the known values, like \(118,098 = 2 \, (3^{n-1})\).

To solve for \(n\):
1. Divide both sides by the first term to isolate the geometric part: \[3^{n-1} = \frac{118,098}{2} = 59,049\]2. Recognize the result as a power of the common ratio: \[59,049 = 3^{10}\]3. Equate the exponents, as the bases are the same, \(n-1 = 10\).4. Solve for \(n\): \[n = 10 + 1 = 11\].
This method helps us conclude that the 11th term of the sequence is 118,098.