Problem 32 - Precalculus: Mathematics for Calculus

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

The common ratio is \(\sqrt{2}\), the fifth term is 4, and the \(n\)th term is \((\sqrt{2})^{n-1}\).

Step 1: Identify the Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by 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 2: Calculate the Fifth Term

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

Step 3: Derive the Formula for the \(n\)th Term

The formula for the \(n\)th term in a geometric sequence is:\[ a_n = a_1 \times r^{n-1} \]In this sequence, \(a_1 = 1\) and \(r = \sqrt{2}\). Substituting these values, we get:\[ a_n = 1 \times (\sqrt{2})^{n-1} = (\sqrt{2})^{n-1}\]

Key Concepts

Understanding the Common Ratio

The common ratio in a geometric sequence is a key characteristic that defines how the sequence progresses from one term to the next. It is the constant factor by which each term in the sequence is multiplied to obtain the subsequent term. To find the common ratio, use the formula:

• Divide any term in the sequence by the preceding term.

In our given sequence: 1, \( \sqrt{2} \), 2, 2\( \sqrt{2} \), identify the common ratio by dividing the second term by the first:

• \( r = \frac{\sqrt{2}}{1} = \sqrt{2} \)

Therefore, \( \sqrt{2} \) is our common ratio, confirming that each subsequent term is simply \( \sqrt{2} \) times the previous term.

Exploring the nth Term Formula

The nth term formula is crucial when dealing with geometric sequences. It allows you to find any term in the sequence without listing all the terms, saving time and effort. The general formula for the n-th term of a geometric sequence is:

• \( a_n = a_1 \times r^{n-1} \)

Here:

• \( a_1 \) represents the first term of the sequence,
• \( r \) is the common ratio,
• \( n \) is the term number you want to find.

In our sequence:

• The first term \( a_1 \) is 1.
• Our common ratio \( r \) is \( \sqrt{2} \).

Thus, the formula to find any term is \( a_n = (\sqrt{2})^{n-1} \). With this, you can plug in any number for \( n \) to easily find the desired term in the sequence.

Calculating the Fifth Term

To calculate the fifth term in a geometric sequence, you use the nth term formula. This involves substituting known values into the formula to directly compute the term you are interested in. For the given sequence with:

• First term \( a_1 = 1 \),
• Common ratio \( r = \sqrt{2} \).

We want to find the fifth term, or when \( n = 5 \). Plug these values into the nth term formula:

• \( a_5 = 1 \times (\sqrt{2})^{5-1} = (\sqrt{2})^4 \).

Since \( (\sqrt{2})^2 = 2 \), it follows that:

• \( (\sqrt{2})^4 = ((\sqrt{2})^2)^2 = 2^2 = 4 \).

Thus, the fifth term is 4, showing how efficiently the nth term formula allows calculation of specific terms within a sequence.