Problem 23 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 2,6,18,54, \dots $$

Step-by-Step Solution

Answer

Common ratio is 3, fifth term is 162, nth term formula is \( a_n = 2 \cdot 3^{n-1} \).

Step 1: Identify the Common Ratio

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio. To find it, divide the second term by the first term: \( r = \frac{6}{2} = 3 \). Thus, the common ratio \( r \) is 3.

Step 2: Calculate the Fifth Term

To find the fifth term of the sequence, continue multiplying by the common ratio. The fourth term is given as 54. Calculate the fifth term by multiplying the fourth term by the common ratio: \( 54 \times 3 = 162 \). Therefore, the fifth term is 162.

Step 3: Derive the General Formula for the nth Term

The nth term of a geometric sequence can be found using the formula \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio. Substituting the given values, the formula is: \( a_n = 2 \cdot 3^{n-1} \).

Step 4: Verify the nth Term Formula

Check the formula by applying it to a known term, such as the third term: \( a_3 = 2 \cdot 3^{3-1} = 2 \cdot 9 = 18 \). Since this matches the sequence, the formula is correct.

Key Concepts

Understanding the Common Ratio in Geometric Sequences

In a geometric sequence, the common ratio is a fundamental element that defines the progression of the sequence. It is the constant factor by which you multiply one term to get to the next. Recognizing this ratio is crucial to understanding how the sequence evolves. For instance, in the sequence provided:

• First term: 2
• Second term: 6
• Third term: 18

To find the common ratio, divide the second term by the first term, which gives us \( r = \frac{6}{2} = 3 \). Each term is 3 times its preceding term. This constant multiplication factor is what we call the common ratio.
It's important to note that the common ratio remains consistent throughout the sequence, providing a uniform pattern that helps in predicting future terms.

Discovering the Fifth Term of the Sequence

The fifth term of a geometric sequence can be found by repeatedly applying the common ratio to the terms until you reach the desired term. In our sequence, the first few known terms are 2, 6, 18, and 54.
Starting from the fourth term, which is 54, the fifth term can be calculated by multiplying 54 by the common ratio, 3. Hence, the fifth term is \( 54 \times 3 = 162 \). It illustrates how the sequence grows rapidly due to the multiplication by the common ratio.
With a clear understanding of how to use the common ratio, calculating any further terms becomes straightforward. This understanding ensures you can confidently determine any term of the sequence.

Using the nth Term Formula for Geometric Sequences

The nth term formula is a powerful tool that allows you to determine any term in a geometric sequence without having to list out all previous terms. The formula is generally expressed as:

• \( a_n = a_1 \cdot r^{n-1} \)
• Where \( a_1 \) is the first term and \( r \) is the common ratio.

For the given sequence, the first term \( a_1 \) is 2, and the common ratio \( r \) is 3. Therefore, the formula becomes \( a_n = 2 \cdot 3^{n-1} \).
This formula can be used to find any term in the sequence; for example, to verify the third term, substitute \( n = 3 \) to get \( a_3 = 2 \cdot 3^{3-1} = 18 \). This matches our already known value, proving the formula's accuracy.
Understanding and using this formula empowers you to precisely calculate any term with ease and reliability.