Problem 31 - Beginning and Intermediate Algebra

Question

Find the indicated term of each geometric sequence. $$1,2,4,8, \dots ; a_{12}$$

Step-by-Step Solution

Answer

The 12th term of the geometric sequence \(1, 2, 4, 8, \dots\) is 2048, as calculated using the formula \(a_n = a_1 * r^{(n - 1)}\) where \(a_1 = 1\), \(r = 2\), and \(n = 12\).

Step 1: Identifying the common ratio

Divide the second term by the first term or the third term by the second term to find the common ratio (r) for the geometric sequence. $$r = \frac{2}{1} = 2$$

Step 2: Apply the formula for the nth term of a geometric sequence

Now that we have the common ratio (r = 2) and the first term in the sequence (a1 = 1), we can find the 12th term using the formula for the nth term of a geometric sequence: $$a_n = a_1 * r^{(n- 1)}$$

Step 3: Calculate the 12th term

Plug in the values into the formula and evaluate the expression for the 12th term (a_12): $$a_{12} = 1 * 2^{(12 - 1)} = 1 * 2^{11} = 1 * 2048 = 2048$$

Step 4: State the result

The 12th term in the given geometric sequence is 2048.

Key Concepts

Understanding the Common Ratio

The common ratio is a fundamental concept in geometric sequences. It defines the relationship between consecutive terms in the sequence. You can determine the common ratio by dividing a term by the preceding term. For example, given the sequence 1, 2, 4, 8, ...
the common ratio, often denoted as \( r \), can be calculated as follows:

• Divide the second term by the first term: \( r = \frac{2}{1} = 2 \)
• Alternatively, divide the third term by the second term: \( r = \frac{4}{2} = 2 \)

In this sequence, the common ratio is \( r = 2 \). Each term is simply double the value of the previous term. This ratio stays constant throughout the entire sequence, which is a defining characteristic of geometric sequences.

Using the nth Term Formula

To find any term in a geometric sequence, we use the nth term formula, which is expressed as:
\(a_n = a_1 \times r^{(n-1)}\)
In this formula:

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

For the sequence 1, 2, 4, 8, ..., if you're looking for the 12th term, plug in the values:
\( a_1 = 1 \), \( r = 2 \), and \( n = 12 \). Thus, the 12th term can be calculated using
\( a_{12} = 1 \times 2^{(12-1)} = 1 \times 2^{11} \).
Therefore, the formula helps transform a potentially complex problem into a straightforward calculation.

Sequence Calculation Process

Calculating a specific term in a geometric sequence involves a clear and precise sequence calculation process. Let's break this down step by step:
1. **Identify the common ratio:** Find the constant factor between consecutive terms (as explained previously).
2. **Plug values into the nth term formula:** Using your first term and common ratio, you simply need to know the position of the desired term.

• Example: For finding the 12th term, use \( a_{12} = 1 \times 2^{11} \).

3. **Perform the calculation:** This could involve simple multiplication or use of exponents.
4. **Verify the result:** It's always good practice to recheck your work to ensure it's accurate.
This methodical approach makes it easier to handle a variety of geometric sequence problems.

Exploring Exponential Growth

Exponential growth in the context of geometric sequences is a fascinating concept. Each term in the sequence is a power of the common ratio, leading to rapidly increasing values.
For example, consider the sequence: 1, 2, 4, 8,

• The term values grow significantly as each is multiplied by the common ratio \( r = 2 \).
• This means each term is a result of an exponential function \( r^{n-1} \).

In our case, the sequence jumps from 1 to 2048 by the 12th term, showcasing exponential growth. This pattern is common in many areas such as population growth, compound interest, and more, where initial sizes multiply over consistent intervals. Understanding this helps in recognizing and modeling such growth patterns.