Problem 27 - Precalculus Mathematics for Calculus

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: 3, Fifth term: 162, \(n\)th term formula: \( a_n = 2 \times 3^{n-1} \).

Step 1: Identify the common ratio

A geometric sequence is defined by the property that each term is a constant multiple of the previous term. This constant multiple is known as the common ratio. To find the common ratio, divide the second term by the first term: \( \text{Common ratio} \ (r) = \frac{6}{2} = 3 \).

Step 2: Calculate the fifth term

In a geometric sequence, the nth term is given by the formula: \[ a_n = a_1 \times r^{n-1} \], where \(a_1\) is the first term and \(r\) is the common ratio.For the fifth term \(a_5\):\[ a_5 = 2 \times 3^{4} = 2 \times 81 = 162 \].

Step 3: Find the general formula for the nth term

Using the same formula for the nth term \( a_n = a_1 \times r^{n-1} \), substitute \(a_1 = 2\) and \(r = 3\):\[ a_n = 2 \times 3^{n-1} \].

Key Concepts

Understanding the Common Ratio in a Geometric Sequence

In the realm of geometric sequences, the common ratio is the magic number. It is what turns one term into the next, and it's consistent for the entire sequence. Think of it as the multiplier that keeps the pattern going.
To find this common ratio, take any term in the sequence (after the first) and divide it by the previous term. For our sequence, 2, 6, 18, and 54, the common ratio (\(r\)) is divided by 6 (second term) by 2 (first term), giving us \(r=3\).
This means every term is three times the one before it. Pretty neat, right? This simple division tells us the core characteristic of the entire sequence. Furthermore, it's critical in understanding how the sequence progresses and helps us calculate any term we want.

Exploring the Nth Term Formula

The nth term formula is like a shortcut for finding any term in a geometric sequence without having to list all the previous terms. It's given by \(a_n = a_1 \times r^{n-1}\). Here, \(a_n\) is the term you want to find, \(a_1\) is the first term, and \(r\) is the common ratio.

Using our sequence, we have a first term \(a_1\) of 2 and a common ratio \(r\) of 3. Substitute these values in, and you get \(a_n = 2 \times 3^{n-1}\). This formula is powerful! It tells you exactly what the nth term is without computing all previous terms, which is especially handy for large sequences.

• Always remember: the nth term formula outlines the entire sequence in one expression.
• It's crucial for predictions and analyses in mathematical modeling.

Calculating the Fifth Term

Calculating specific terms in a geometric sequence is straightforward once you know the formula for the nth term. For the fifth term, we simply insert 5 for \(n\) into our formula \(a_n = 2 \times 3^{n-1}\).
Let's calculate step by step:

• Replace \(n\) with 5: \(a_5 = 2 \times 3^{5-1}\)
• Simplify the exponent: \(3^4 = 81\)
• Complete the multiplication: \(a_5 = 2 \times 81 = 162\)

The fifth term in the sequence is hence 162, illustrating how the sequence rapidly increases as it progresses. Understanding this helps in grasping the exponential nature of geometric sequences.