Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=8, r=-\frac{3}{4}, 9\)th term

Step-by-Step Solution

Answer

After the above steps, the 9th term of the given geometric sequence is found to be \(a_{9}\) both by using table feature of a graphing utility and algebraically.

Step 1: Using the Table Feature of a Graphing Utility

Using a graphing utility, plot an equation of the form \(y = a_{1} * r^{(n-1)}\), where \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) are the terms of the sequence. Substitute \(a_{1} = 8\) and \(r=-3/4\) in the formula and plot the sequence. Then navigate to the 9th term.

Step 2: Identify the 9th Term Using the Table Feature

The graphing utility plotted in Step 1 would give a table with the values of the terms in the geometric sequence. Locate the value of the 9th term in the table. This is the required solution using the table feature of a graphing utility.

Step 3: Using the Geometric Sequence Formula Algebraically

To find the 9th term algebraically, use the formula for any term of a geometric sequence: \(a_{n} = a_{1} * r^{(n-1)}\). Substitute the given first term \(a_{1} = 8\), the common ratio \(r=-3/4\) and \(n = 9\) into the formula.

Step 4: Calculate the 9th Term Algebraically

After substituting the given values into the geometric sequence formula in Step 3, perform the calculation \(a_{9} = 8 * (-3/4)^8\).

Step 5: Simplify the Result

Simplify the calculation to get the answer.

Key Concepts

Graphing Utility

A graphing utility is a helpful tool for visualizing mathematical sequences like geometric sequences. It allows you to see the progression of the sequence by plotting its terms. In this exercise, you use it to find a specific term within a geometric sequence. Here's a simple way to use a graphing utility to discover the 9th term:

• First, input the sequence formula, which is typically in the form of \(y = a_{1} \times r^{(n-1)}\), where \(a_{1}\) is the starting term and \(r\) is the common ratio.
• With the initial term \(a_{1} = 8\) and the common ratio \(r = -\frac{3}{4}\), input these values into your graphing calculator.
• As the graph plots the sequence, use the table function to navigate to the 9th term.

Using the graphing utility not only simplifies finding specific terms but also provides a visual representation of the sequence's behavior.

Sequence Formula

The sequence formula is a crucial concept when dealing with geometric sequences. Specifically, the formula for a geometric sequence is \(a_{n} = a_{1} \times r^{(n-1)}\). This formula helps us find the \(n\)th term of the sequence. Here's a breakdown of the formula and its components:

• \(a_{n}\) represents the term in the sequence we want to calculate.
• \(a_{1}\) is the first term given in the sequence. For this problem, it is 8.
• \(r\) is known as the common ratio; it's the factor by which we multiply each term to get the next term, and in this case, it's \(-\frac{3}{4}\).
• \((n-1)\) is the exponent, indicating the number of times the common ratio is multiplied to escalate from the first term to the \(n\)th term.

Knowing how to manipulate and use this formula is key to solving problems involving geometric sequences.

Common Ratio

The common ratio is a fundamental element in geometric sequences. It is the factor by which each term in the sequence is multiplied to obtain the next term. Here are some essential points to understand:

• The common ratio \(r\) can be any real number: positive, negative, a fraction, or an integer.
• In our example, the common ratio is \(-\frac{3}{4}\), which means each term is obtained by multiplying the previous term by \(-\frac{3}{4}\).
• If \(|r| < 1\), the terms of the sequence will decrease in magnitude.
• If \(|r| > 1\), the terms will grow larger in magnitude.

Understanding the common ratio helps in forecasting the behavior and progression of a geometric sequence, whether it'll converge to zero or increase indefinitely.

Term Calculation

Calculating a specific term in a geometric sequence involves using the sequence formula effectively. In the provided exercise, you were tasked to find the 9th term in a sequence where the first term \(a_{1} = 8\) and the common ratio \(r = -\frac{3}{4}\).Here's how to calculate that 9th term step-by-step:

• Substitute \(a_{1} = 8\), \(r = -\frac{3}{4}\), and \(n = 9\) into the geometric sequence formula: \(a_{n} = a_{1} \times r^{(n-1)}\).
• This gives us \(a_{9} = 8 \times \left(-\frac{3}{4}\right)^{8}\).
• Calculate \((-\frac{3}{4})^{8}\). This involves multiplying \(-\frac{3}{4}\) by itself eight times.
• Multiply the result by 8 to get the value of the 9th term.

Calculating terms algebraically ensures you understand the relationship between each component of the formula and helps verify any calculator results.