Problem 30 - 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

Common ratio: \( \sqrt{2} \); fifth term: 4; general \( n \)-th term: \( (\sqrt{2})^{n-1} \).

Step 1: Identify the Common Ratio

To find the common ratio of the geometric sequence, we divide the second term by the first term: \( \frac{\sqrt{2}}{1} = \sqrt{2} \). Verify this ratio by ensuring that \( \frac{2}{\sqrt{2}} = \sqrt{2} \) and \( \frac{2\sqrt{2}}{2} = \sqrt{2} \). Thus, the common ratio \( r = \sqrt{2} \).

Step 2: Calculate the Fifth Term

Using the formula for the \( n \)-th term of a geometric sequence, \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 = 1 \) and \( r = \sqrt{2} \), we compute the fifth term: \( a_5 = 1 \times (\sqrt{2})^{4} = 1 \times (2)^{2} = 4 \).

Step 3: Find the General \(n\)-th Term Formula

Using the same formula \( a_n = a_1 \cdot r^{n-1} \), substitute \( a_1 = 1 \) and \( r = \sqrt{2} \) to find the general expression: \( a_n = 1 \cdot (\sqrt{2})^{n-1} = (\sqrt{2})^{n-1} \). This formula gives the \( n \)-th term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key factor that determines how each term relates to the next. It is essentially the constant factor you multiply the current term with to get to the next term in the sequence. To find the common ratio, you can divide any term by the previous term.

For instance, in the sequence given:

• The first term is 1.
• The second term is \( \sqrt{2} \).
• Divide the second term by the first: \( \frac{\sqrt{2}}{1} = \sqrt{2} \).

This calculation yields the common ratio of \( \sqrt{2} \). It is important to verify this by checking other consecutive terms. Indeed, dividing the third term 2 by the second term \( \sqrt{2} \), \( \frac{2}{\sqrt{2}} = \sqrt{2} \), confirms it. Also, dividing the fourth term \( 2 \sqrt{2} \) by the third term 2 confirms that the common ratio remains consistent across the sequence. Correct identification of the common ratio is essential as it forms the basis for finding other elements in the sequence.

Nth Term Formula

The nth term formula in a geometric sequence helps you determine any term's value in the sequence without listing all previous terms. To calculate the nth term, you use the formula: \[ a_n = a_1 \cdot r^{n-1} \] where:

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

In our sequence, the first term \( a_1 = 1 \), and the common ratio \( r = \sqrt{2} \). Plug these values into the formula to get the nth term: \( a_n = 1 \cdot (\sqrt{2})^{n-1} \).

This formula provides a simple way to find any term in the geometric sequence without having to compute each previous term. It greatly simplifies processes in mathematics requiring sequence terms, making calculations quicker and more manageable.

Fifth Term

To find the fifth term of a geometric sequence, we leverage the nth term formula: \( a_n = a_1 \cdot r^{n-1} \).

• For the fifth term, let \( n = 5 \).
• Given: \( a_1 = 1 \) and the common ratio \( r = \sqrt{2} \).
• Plug into the formula: \( a_5 = 1 \cdot (\sqrt{2})^{4} \).

Calculate \( (\sqrt{2})^4 \):
\[ (\sqrt{2})^4 = (\sqrt{2} \times \sqrt{2}) \times (\sqrt{2} \times \sqrt{2}) = 2 \times 2 = 4 \]Thus, the fifth term \( a_5 \) equals 4. This application of the formula is practical for quickly identifying specific terms within the sequence, especially when dealing with larger values of \( n \) that would otherwise be cumbersome to calculate sequentially.