Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$2,6,18,54, \ldots$$

Step-by-Step Solution

Answer

The nth term is \( a_n = 2 \cdot 3^{n-1} \). The fifth term is 162, and the eighth term is 4374.

Step 1: Identify the First Term (a)

In a geometric sequence, the first term is denoted as \( a \). In this sequence, the first term is \( a = 2 \).

Step 2: Calculate the Common Ratio (r)

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

Step 3: Determine the General Formula for the nth Term

The formula for the \( n \)th term of a geometric sequence is given by \( a_n = a \cdot r^{n-1} \). Substituting the values we have, \( a_n = 2 \cdot 3^{n-1} \).

Step 4: Find the Fifth Term (a_5)

Substitute \( n = 5 \) into the general formula \( a_n = 2 \cdot 3^{n-1} \): \( a_5 = 2 \cdot 3^{5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162 \).

Step 5: Find the Eighth Term (a_8)

Substitute \( n = 8 \) into the general formula \( a_n = 2 \cdot 3^{n-1} \): \( a_8 = 2 \cdot 3^{8-1} = 2 \cdot 3^7 = 2 \cdot 2187 = 4374 \).

Key Concepts

Common Ratio

In a geometric sequence, one of the key aspects that defines the sequence is the "common ratio", often represented by the symbol \( r \). The common ratio is a constant factor between consecutive terms of the sequence. To find this ratio, you divide any term by the previous term.

• For example, in the sequence given: \( 2, 6, 18, 54, \ldots \), we can find the common ratio by dividing the second term \( 6 \) by the first term \( 2 \).
• Thus, \( r = \frac{6}{2} = 3 \).

New lines in the sequence are generated by multiplying the previous term by the common ratio. Understanding the common ratio is essential for any calculations involving geometric sequences, such as finding the nth term or summing terms in a geometric series.

Nth Term Formula

The formula for finding the nth term in a geometric sequence is a very handy tool. It allows us to determine any term in the sequence without listing all the preceding terms. The formula is:

• \( a_n = a \cdot r^{n-1} \)

Here, \( a \) represents the first term in the sequence, \( r \) is the common ratio, and \( n \) is the term number you are looking for.
For the sequence \( 2, 6, 18, 54, \ldots \), the first term \( a = 2 \) and the common ratio \( r = 3 \). Thus, the nth term formula becomes \( a_n = 2 \cdot 3^{n-1} \).
Using this formula:

• To find the fifth term \( a_5 \), substitute \( n = 5 \) into the formula: \( a_5 = 2 \cdot 3^{5-1} = 2 \cdot 81 = 162 \).
• To find the eighth term \( a_8 \), substitute \( n = 8 \): \( a_8 = 2 \cdot 3^{8-1} = 2 \cdot 2187 = 4374 \).

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. Understanding this concept is beneficial when you need to calculate the cumulative total of a sequence's terms, up to a certain point. For a finite geometric series, the sum \( S_n \) of the first \( n \) terms is calculated by the formula:

• \( S_n = a \frac{1 - r^n}{1 - r} \)

where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. This formula works when \( r eq 1 \).
For our sequence \( 2, 6, 18, 54, \ldots \) with \( a = 2 \) and \( r = 3 \), if you wanted to find the sum of the first five terms, you would use the formula:

• \( S_5 = 2 \cdot \frac{1 - 3^5}{1 - 3} = 2 \cdot \frac{1 - 243}{-2} = 2 \cdot 121 = 242 \).

Understanding geometric series helps you gain insights into the cumulative nature of sequences and can be particularly useful in financial calculations and computer science simulations.