Problem 26 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 1, \sqrt{2}, 2,2 \sqrt{2}, \ldots $$

Step-by-Step Solution

Answer

Common ratio is \(\sqrt{2}\), fifth term is 4, \(n\)th term is \(a_n = 1 \times (\sqrt{2})^{n-1}\).

Step 1: Identify the First Term

The first term of the sequence is the value present at the beginning. Here, the first term \(a_1\) is 1.

Step 2: Calculate the Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{\sqrt{2}}{1} = \sqrt{2}\).

Step 3: Find the Fifth Term

To find the fifth term \(a_5\), use the formula for the \(n\)th term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\). Substitute the known values: \(a_5 = 1 \times (\sqrt{2})^{5-1} = 1 \times (\sqrt{2})^4 = 4\).

Step 4: Derive the Expression for the nth Term

Use the formula for the \(n\)th term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\). Substitute the known values: \(a_n = 1 \times (\sqrt{2})^{n-1}\).

Key Concepts

Understanding the Common Ratio

The common ratio in a geometric sequence is crucial since it defines how we progress from one term to another. Simply put, it's the number we multiply each term by to get the next one. This ratio remains consistent throughout the sequence. In our example, the geometric sequence starts with the numbers 1 and \( \sqrt{2} \). To find the common ratio \( r \), you take the second term and divide it by the first term:

• Second term: \( \sqrt{2} \)
• First term: 1

So, \( r = \frac{\sqrt{2}}{1} = \sqrt{2} \). Every subsequent term in this sequence is simply the previous term multiplied by \( \sqrt{2} \). This consistency makes it easier to predict or calculate any term in the sequence as long as you know the previous one.

Deriving the nth Term Formula

The \( n \)th term formula in a geometric sequence allows us to find any term in the sequence without having to list all the previous terms. This formula is incredibly useful for large sequences or when you need to quickly determine distant terms. The formula for the \( n \)th term \( a_n \) is given by:

• \( a_n = a_1 \times r^{n-1} \)

Where:

• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term in the sequence.

In the example sequence, where the first term \( a_1 \) is 1 and the common ratio is \( \sqrt{2} \), the formula becomes:\[ a_n = 1 \times (\sqrt{2})^{n-1} \]This formula can now be used to calculate any term in the sequence directly by plugging in the desired position \( n \). It's efficient and saves time, especially for terms that are far into the sequence.

Calculating the Fifth Term

To find a specific term in the sequence, like the fifth term, you can use the formula for the \( n \)th term. As derived before:\[ a_n = a_1 \times r^{n-1} \]For the fifth term, set \( n = 5 \). We already know:

• \( a_1 = 1 \)
• \( r = \sqrt{2} \)

Substitute these values into the formula:\[ a_5 = 1 \times (\sqrt{2})^{5-1} \]This simplifies to:\[ a_5 = 1 \times (\sqrt{2})^4 = 4 \]Thus, the fifth term in this geometric sequence is 4. This method shows the power of the \( n \)th term formula, making it easy to find specific terms in the sequence without calculating all the intermediate terms.