Problem 30 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the nth 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, nth term: \((\sqrt{2})^{n-1}\).

Step 1: Identify the Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio. Let's find the common ratio, \( r \), by dividing the second term by the first term: \( r = \frac{\sqrt{2}}{1} = \sqrt{2} \). Now, confirm that this ratio works for the next pair of terms as well by dividing the third term by the second term: \( \frac{2}{\sqrt{2}} = \sqrt{2} \). Therefore, the common ratio is \( r = \sqrt{2} \).

Step 2: Calculate the Fifth Term

Using the common ratio \( \sqrt{2} \) and knowing that the first term \( a_1 \) is 1, we find the fifth term of the sequence. The nth term of a geometric sequence is given by \( a_n = a_1 \cdot r^{n-1} \). For the fifth term \( n = 5 \): \[ a_5 = 1 \cdot (\sqrt{2})^{5-1} = (\sqrt{2})^4 = (\sqrt{2}^2)^2 = 2^2 = 4 \].

Step 3: Derive the General Formula for the nth Term

To determine the nth term of the sequence, use the formula for the nth term of a geometric sequence: \( a_n = a_1 \cdot r^{n-1} \). Here, \( a_1 = 1 \) and \( r = \sqrt{2} \). Thus, the formula for the nth term is: \[ a_n = 1 \cdot (\sqrt{2})^{n-1} = (\sqrt{2})^{n-1} \].

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element. It is the number we multiply each term by to reach the next term in the sequence.
Think of it as a sort of "growth factor" for the sequence. In our exercise, we need to identify the common ratio for the sequence 1, \( \sqrt{2} \), 2, 2\( \sqrt{2} \), ... .
To find the common ratio:

• Divide the second term by the first term. For this sequence: \( r = \frac{\sqrt{2}}{1} = \sqrt{2} \).
• Verify it by dividing the third term by the second term: \( \frac{2}{\sqrt{2}} = \sqrt{2} \).

This common ratio works consistently for the sequence, confirming that \( r = \sqrt{2} \).
Every time you have a geometric sequence problem, remember - the common ratio is your key to understanding how the sequence behaves.

nth Term Formula

The nth term formula is your blueprint for finding any term in a geometric sequence without listing out all the previous terms.
The formula is given by:\[ a_n = a_1 \cdot r^{n-1} \]Let's break this down:

• \( a_n \) is the term you're trying to find.
• \( a_1 \) is the first term of the sequence, which in this case is 1.
• \( r \) is the common ratio, which we found to be \( \sqrt{2} \).
\( n \) is the position of the term in the sequence.

With our sequence, the nth term formula simplifies to:\[ a_n = (\sqrt{2})^{n-1} \]This shows how each term builds upon the previous one by scaling it with the common ratio raised to the power of one less than the position number.

Geometric Progression

A geometric progression—or sequence—is a series of numbers where each term is derived from the previous one by multiplication by a fixed, non-zero number called the common ratio. Such sequences can grow very quickly!
The sequence 1, \( \sqrt{2} \), 2, 2\( \sqrt{2} \), ... is a perfect example. Here are its critical features:

• The starting point, or first term (\( a_1 \)), is 1.
• The methodology, a consistent multiplier or common ratio, is \( \sqrt{2} \).
• The terms follow a precise, exponential growth pattern due to the multiplication by \( \sqrt{2} \).

Understanding geometric progressions unlocks the door to mastering various mathematical concepts and calculations connected to sequences and series.

Sequence Terms Calculation

Calculating specific terms in a geometric sequence is straightforward once you've got the common ratio and nth term formula.
For instance, let's calculate the fifth term in our sequence using the formula:\[ a_5 = (\sqrt{2})^{5-1} \]This becomes:

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

Thus, the fifth term is 4.
With this new understanding, you can quickly compute any term in the sequence by applying the common ratio and plugging into the nth term formula.