Problem 26 - Precalculus: Mathematical 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 Initial Terms

First, identify the initial terms provided in the sequence: 1, \(\sqrt{2}\), 2, \(2\sqrt{2}\).

Step 2: Determine the Common Ratio

The common ratio \(r\) is found by dividing the second term by the first term: \(r = \frac{\sqrt{2}}{1} = \sqrt{2}\). Verify by checking subsequent terms: \(\frac{2}{\sqrt{2}} = \sqrt{2}\) and \(\frac{2\sqrt{2}}{2} = \sqrt{2}\). Therefore, the common ratio is consistent at \(\sqrt{2}\).

Step 3: Find the Fifth Term

The fifth term can be calculated using the formula for the \(n\)-th term of a geometric sequence: \(a_n = a_1 \cdot r^{n-1}\). Substitute the known values to find the fifth term: \(a_5 = 1 \cdot (\sqrt{2})^{4} = 1 \cdot 4 = 4\).

Step 4: General Formula for the \(n\)-th Term

The general formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\). Substitute \(a_1 = 1\) and \(r = \sqrt{2}\) into the formula: \(a_n = 1 \cdot (\sqrt{2})^{n-1} = (\sqrt{2})^{n-1}\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element. It's what makes the sequence "geometric," meaning that each term after the first is the product of the previous term and the common ratio.
To find the common ratio, you divide any term by the term preceding it. In our example, the sequence given is: 1, \(\sqrt{2}\), 2, \(2\sqrt{2}\) and so on.
Let's calculate the common ratio \(r\):

• Take the second term \(\sqrt{2}\) and divide it by the first term 1: \(r = \frac{\sqrt{2}}{1} = \sqrt{2}\).
• Verify the common ratio is consistent for other terms just in case; \(\frac{2}{\sqrt{2}} = \sqrt{2}\) and \( \frac{2\sqrt{2}}{2} = \sqrt{2} \).

This checks out for all consecutive terms, confirming that our common ratio is indeed \(\sqrt{2}\). By determining \(r\) this way, you ensure the sequence follows the pattern dictated by a geometric progression.

Nth Term Formula

The nth term formula is like the key that unlocks any position within a geometric sequence. This formula allows you to calculate any term without having to manually compute all the preceding terms.
The formula is:

• \(a_n = a_1 \cdot r^{n-1}\)

Where:

• \(a_n\) is the nth term,
• \(a_1\) is the first term,
• \(r\) is the common ratio,
• \(n\) is the term number you're trying to find.

In our example:

• \(a_1 = 1\)
• \(r = \sqrt{2}\)
• The nth term formula becomes \(a_n = (\sqrt{2})^{n-1}\).

This means for any term number \(n\), you can simply plug \(n\) into the formula to find \(a_n\). For instance, to find the fifth term, you substitute \(n = 5\) to get: \(a_5 = (\sqrt{2})^{4} = 4.\)

Geometric Progression

A geometric progression (or sequence) is a series of numbers where each term after the first is found by multiplying the previous one by a constant, known as the common ratio. It's like you're stepping through the sequence by multiplying each step.
Here’s how you can recognize a geometric progression:

• The ratio \(r\) between any term and the term before it is always the same.
• You can generate the sequence by repeatedly multiplying the common ratio by each subsequent term.

In the sequence: 1, \(\sqrt{2}\), 2, \(2\sqrt{2}\), the common ratio \(r = \sqrt{2}\) is this "stepping" multiplier.
Thus, you start with 1 and each term thereafter is the product of \(\sqrt{2}\), showing a consistent geometric pattern.
Geometric progressions showcase exponential growth or decay, making them foundational in understanding patterns and trends in mathematics.