Problem 31 - Algebra 2 and Trigonometry

Question

In a geometric sequence, \(a_{1}=2\) and \(a_{3}=16 .\) Find \(a_{6}\)

Step-by-Step Solution

Answer

The sixth term \( a_6 \) is \( 256\sqrt{2} \).

Step 1: Understand the Problem

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio, denoted as \( r \). We are given the first term \( a_1 = 2 \) and the third term \( a_3 = 16 \). We need to find the sixth term \( a_6 \).

Step 2: Find the Common Ratio

In a geometric sequence, the \( n \)-th term is given by \( a_n = a_1 \cdot r^{n-1} \). Since we know \( a_1 = 2 \) and \( a_3 = 16 \), we can use the formula for \( a_3 \):\[ a_3 = 2 \cdot r^2 = 16 \]Solve for \( r \):\[ r^2 = \frac{16}{2} = 8 \]\[ r = \sqrt{8} = 2\sqrt{2} \]

Step 3: Calculate the Sixth Term

Now that we have the common ratio \( r = 2\sqrt{2} \), use it to find \( a_6 \). The formula for the \( n \)-th term is \( a_n = a_1 \cdot r^{n-1} \), so:\[ a_6 = 2 \cdot (2\sqrt{2})^{5} \]Calculate \( (2\sqrt{2})^5 \):\[ (2\sqrt{2})^5 = 2^5 \cdot (\sqrt{2})^5 = 32 \cdot 4\sqrt{2} = 128\sqrt{2} \]Finally, we find \( a_6 \):\[ a_6 = 2 \times 128\sqrt{2} = 256\sqrt{2} \]

Step 4: Conclusion

The sixth term of the geometric sequence is \( a_6 = 256\sqrt{2} \).

Key Concepts

Common Ratio

In a geometric sequence, the "common ratio" is the constant factor between consecutive terms. This means each term after the first is obtained by multiplying the previous term by this fixed number, often denoted as \( r \).
Understanding the common ratio is crucial because it reveals the pattern that the sequence follows.
For example, suppose the first term \( a_1 \) is given as 2 and the third term \( a_3 \) is 16. To find this all-important factor, you can use the formula for the \( n \)-th term in a geometric sequence, \( a_n = a_1 \cdot r^{n-1} \). By substituting the known values, \( a_3 = 2 \cdot r^2 = 16 \), you solve for \( r \), giving \( r^2 = 8 \), which simplifies to \( r = 2\sqrt{2} \).

• Determine the pattern or growth by finding \( r \).
• Multiply or divide terms to verify the common ratio.

Recognizing the common ratio allows you to predict future terms and ties directly into calculating other aspects of a sequence.

Nth Term Formula

The "nth term formula" is a powerful tool that allows you to determine any term in a geometric sequence without listing all previous terms. This formula is represented by \( a_n = a_1 \cdot r^{n-1} \) where:

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

To illustrate, if you want to find the sixth term of a sequence where \( a_1 = 2 \) and \( r = 2\sqrt{2} \), you substitute these values into the formula: \[ a_6 = 2 \cdot (2\sqrt{2})^5 \]Calculate \((2\sqrt{2})^5\) which results in a number that you then multiply by \(a_1\), leading to \( a_6 = 256\sqrt{2} \). The nth term formula offers a shortcut to get to any point in the sequence quickly and accurately. This is especially useful in sequences with a large number of terms.

Geometric Progression

A "geometric progression" is a sequence where each term after the first is the product of the previous term and a fixed, non-zero number known as the common ratio. This progression is different from an arithmetic progression, where terms increase by a constant sum. Instead, a geometric progression grows (or shrinks) exponentially.
Consider our exercise with the series beginning with 2, and the common ratio as \(2\sqrt{2}\). This tells us:

• Each subsequent term is the previous term times \(2\sqrt{2}\).
• The series grows quite rapidly, especially as \( n \) (term position) becomes large.

This kind of progression characterizes many real-world phenomena, such as population growth, financial interest calculations, and certain types of waves. Understanding the concept of geometric progression helps you recognize patterns and predict future values in various practical and theoretical contexts.