Problem 37 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Sequence

This is a geometric sequence where the first term is \(a = 2\) and the common ratio \(r\) can be calculated from the second term: \(r = \frac{6}{2} = 3\). Thus, the sequence follows a pattern where each term is 3 times the previous term.

Step 2: Use the Geometric Sequence Formula

The formula for the \(n\)-th term of a geometric sequence is \(a_n = a \cdot r^{n-1}\). In this sequence, \(a = 2\) and \(r = 3\), so the formula becomes \(a_n = 2 \cdot 3^{n-1}\).

Step 3: Set Up the Equation

We need to find \(n\) such that \(a_n = 118,098\). Thus, the equation to solve is \(2 \cdot 3^{n-1} = 118,098\).

Step 4: Solve the Equation

First, divide both sides by 2: \[3^{n-1} = \frac{118,098}{2} = 59,049.\]Next, recognize 59,049 as a power of 3. Since \(3^{10} = 59,049\), \(n-1 = 10\). Therefore, \(n = 11\).

Step 5: Verify the Solution

Verify by substituting \(n = 11\) back into the sequence formula. Calculate \[a_{11} = 2 \cdot 3^{10} = 2 \cdot 59,049 = 118,098.\]The calculations confirm that the 11th term is indeed 118,098.

Key Concepts

Sequence Formula

In a geometric sequence, each term is generated by multiplying the previous term by a constant called the 'common ratio'. A geometric sequence formula helps us find any term in the sequence without writing all the terms up to that point.

For a geometric sequence, the formula for the n-th term, often represented as \( a_n \), is:

• \( a_n = a \cdot r^{n-1} \)

Here, \( a \) is the first term and \( r \) is the common ratio. The power \( n-1 \) represents the position of the term within the sequence.

Using the formula, you can jump directly to any term in the sequence without computing all the previous terms. For example, if the first term of a sequence is 2, and the common ratio is 3, the sequence formula \( a_n = 2 \cdot 3^{n-1} \) helps you find any term directly.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get to the next term. It is denoted by \( r \).

To find the common ratio, divide the second term by the first term. In our example sequence \(2, 6, 18, \ldots \), the common ratio is:

• \( r = \frac{6}{2} = 3 \)

This means every term is 3 times the previous term. The common ratio is crucial because it helps to define how the sequence progresses and is used in the geometric sequence formula.

To identify whether a sequence is geometric, check if the common ratio between all successive terms is the same.

n-th Term

The n-th term in a sequence represents a term located at the position \( n \). In a geometric sequence, calculating the n-th term is straightforward with the sequence formula.

Given a first term \( a \) and a common ratio \( r \), the n-th term formula is:

• \( a_n = a \cdot r^{n-1} \)

Let's see how this formula works through our example problem. We need to find which term equals 118,098 in the sequence \(2, 6, 18, \ldots \). Set up the equation:

• \( 2 \cdot 3^{n-1} = 118,098 \)

By solving this equation, you find that the power \( n-1 = 10 \) (since \( 3^{10} = 59,049 \)), leading to \( n = 11 \). Hence, the 11th term in this sequence is 118,098.

Calculating the n-th term helps in understanding the growth and pattern of the sequence seamlessly.