Problem 15 - Algebra for College Students

Question

\text { The } 11 \text { th term of } 768,384,192,96, \ldots \quad \frac{3}{4}

Step-by-Step Solution

Answer

The 11th term is \( \frac{3}{4} \).

Step 1: Identify the pattern or sequence type

We have the sequence 768, 384, 192, 96, ... . It is helpful to identify if this sequence follows an arithmetic, geometric, or any other specific pattern. Looking at the given numbers, let's check the factor by which each term is divided from the previous one.

Step 2: Determine the common ratio

Divide the second term by the first term: \( \frac{384}{768} = \frac{1}{2} \). Do the same for the next pair: \( \frac{192}{384} = \frac{1}{2} \) and \( \frac{96}{192} = \frac{1}{2} \). Thus, this is a geometric sequence with a common ratio of \( \frac{1}{2} \).

Step 3: Write the formula for the nth term

For a geometric sequence, the nth term is found using the formula: \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio. In this case, \( a_1 = 768 \) and \( r = \frac{1}{2} \).

Step 4: Calculate the 11th term

Substitute \( n = 11 \) into our formula: \[ a_{11} = 768 \cdot \left(\frac{1}{2}\right)^{(11-1)} \]Simplify the exponent and calculate:\[ a_{11} = 768 \cdot \left(\frac{1}{2}\right)^{10} \] \[ = 768 \cdot \left(\frac{1}{1024}\right) \]\[ = \frac{768}{1024} = \frac{3}{4} \]

Key Concepts

Common Ratio

In mathematics, a geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio". This common ratio is crucial because it tells us how our sequence progresses, either increasing or decreasing.

To find the common ratio, you divide any term in the sequence by the previous term. This is a straightforward step that determines the relationship between the terms. For example, with the sequence 768, 384, 192, 96, ..., you divide the second term by the first term:

• \( rac{384}{768} = rac{1}{2} \)

Repeat this process for the following terms:

• \( rac{192}{384} = rac{1}{2} \)
• \( rac{96}{192} = rac{1}{2} \)

Each result gives us the same common ratio of \( rac{1}{2} \).

This consistent ratio confirms that we are dealing with a geometric sequence, where each term is half the previous term. Understanding the common ratio is essential for identifying the sequence type and solving for any term in the sequence.

Nth Term Formula

After confirming the common ratio, the next step in working with a geometric sequence is to use the nth term formula. This formula allows us to find any term in the sequence without listing all the previous ones. The formula for the nth term, \( a_n \), is given by:

\[ a_n = a_1 \cdot r^{(n-1)} \]

Here, \( a_1 \) represents the first term of the sequence, \( r \) is the common ratio, and \( n \) indicates the term number we want to find. For our sequence 768, 384, 192, 96, ..., we know:

• First term, \( a_1 = 768 \)
• Common ratio, \( r = \frac{1}{2} \)

To find the 11th term, substitute the appropriate values into the formula:

\[ a_{11} = 768 \cdot \left(\frac{1}{2}\right)^{(11-1)} \]

Calculating this expression provides the 11th term. This method beautifully demonstrates the power of mathematical formulas to simplify complex calculations into manageable steps.

Sequence Pattern Identification

Identifying the type of sequence you are working with is the first and arguably most important step in solving sequence problems. With various types of sequences such as arithmetic, geometric, and others, recognizing the pattern early on saves time and effort.

For our sequence 768, 384, 192, 96, ..., determining whether it's arithmetic or geometric requires examining how each term relates to the next. By calculating ratios like:

• From 768 to 384, which leaves \( \frac{1}{2} \)
• From 384 to 192, again \( \frac{1}{2} \)

Seeing that the ratio is consistent, we're dealing with a geometric sequence. Identifying sequences accurately is foundational for solving any related exercises and applying the correct formulas. Understanding the sequence type helps you predict future terms and solve for specific term values effectively.