Problem 13 - Algebra for College Students

Question

Find the required term for each geo- metric sequence. The 8 th term of \(\frac{1}{2}, 1,2,4, \ldots\) 64

Step-by-Step Solution

Answer

The 8th term of the sequence is 64.

Step 1: Identify the Common Ratio

In a geometric sequence, the ratio between consecutive terms is constant. To find this ratio, divide the second term by the first term. For the sequence \(\frac{1}{2}, 1, 2, 4, \ldots\), the common ratio \(r\) is given by: \[ r = \frac{1}{2} \times 2 = 2 \]

Step 2: Write the Formula for the n-th Term of the Sequence

The n-th term of a geometric sequence is given by the formula \( a_n = a_1 \cdot r^{(n-1)} \), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 3: Substitute the Known Values into the Formula

Given \(a_1 = \frac{1}{2}\) and \(r = 2\), we're finding the 8th term, thus \(n = 8\). Substitute these values into the formula: \[ a_8 = \frac{1}{2} \cdot 2^{7} \]

Step 4: Calculate \(2^{7}\)

Calculate the power of 2: \[ 2^{7} = 128 \]

Step 5: Calculate the 8th Term

Substitute the value of \(2^{7}\) back into the equation for the 8th term: \[ a_8 = \frac{1}{2} \cdot 128 = 64 \]

Step 6: Conclusion

The 8th term of the sequence \(\frac{1}{2}, 1, 2, 4, \ldots\) is 64.

Key Concepts

Common Ratio

In every geometric sequence, there is a consistent value known as the common ratio. This value is defined as the ratio of any term in the sequence to its previous term. For example, consider the sequence \( \frac{1}{2}, 1, 2, 4, \ldots \). To find the common ratio \( r \), take the second term and divide it by the first term, resulting in:

• \( r = \frac{1}{2} \times 2 = 2 \)

The common ratio remains constant across all terms in the sequence. If you check any two consecutive terms, you'll find this ratio unchanged. Knowing the common ratio helps to predict future terms and understand the structure of the sequence.

n-th Term Formula

The n-th term formula is a valuable tool in finding any term in a geometric sequence without the need to list all preceding terms. The formula is as follows:\[ a_n = a_1 \cdot r^{(n-1)} \]Where:

• \( a_n \) is the n-th term of the sequence.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term in the sequence.

Using this formula simplifies the process of finding any term, by allowing a direct calculation based on its position rather than calculating all terms up to that point. Simply plug in the known values, and you'll find the desired term quickly.

Power Calculation

Calculating powers is a key operation in finding terms of a geometric sequence using the n-th term formula. In the formula \( a_n = a_1 \cdot r^{(n-1)} \), the expression \( r^{(n-1)} \) requires the calculation of a power. For instance:If you have \( r = 2 \) and you need to find the 8th term, you calculate:

• \( 2^{7} \)
• Which is \( 128 \)

This step requires understanding of exponentiation, which is repeated multiplication of the base by itself. The number \( 2^{7} \) means multiplying 2 by itself seven times (\( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \)).
Knowing how to perform power calculations efficiently is vital for solving and understanding geometric sequences, especially when located in contexts involving larger numbers or higher exponents.