Problem 84 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=8(-0.75)^{n-1}$$

Step-by-Step Solution

Answer

The first 10 terms of the sequence are: 8, -6, 4.5, -3.375, 2.53125, -1.89844, 1.42383, -1.06787, 0.800654, and -0.600491. These are plotted on the graph using a graphing utility.

Step 1: Understand the Geometric Sequence

A geometric sequence is one where each term is multiplied by a common ratio to get the next term. In the given sequence, \( a_{n} = 8(-0.75)^{n-1} \), -0.75 is the common ratio. The term 8 is the first term of the sequence (when n=1).

Step 2: Calculate the First 10 Terms

Plug values of n=1 to n=10 into the formula to get the first 10 terms of the sequence. These 10 terms will be: 8, -6, 4.5, -3.375, 2.53125, -1.89844, 1.42383, -1.06787, 0.800654, and -0.600491 respectively.

Step 3: Use A Graphing Utility

A good graphing utility can be used to graph the sequence. The x-coordinates will be the terms' numbers (1 to 10), and the y-coordinates will be the terms themselves (the values calculated in step 2).

Key Concepts

Graphing Utility

A graphing utility is an essential tool in visualizing mathematical sequences and functions. This tool allows you to input a mathematical function or sequence, like the geometric sequence in our exercise, and see a visual representation of it on a coordinate grid. By doing this, complex sequences become more understandable and patterns become evident through the graph.

In our specific example, you will use the graphing utility to plot the first 10 terms of the sequence given by the expression \(a_{n} = 8(-0.75)^{n-1}\). Here’s what to do:

• Set up your graph with the x-axis representing the term numbers (from 1 to 10).
• The y-axis will correspond to the value of each term you calculated, such as 8, -6, 4.5, and so on.

Once plotted, the graph will show a visual alternation between positive and negative values, reflecting the negative common ratio. It's a fantastic way to see how the sequence's values are getting closer to zero and changing sign each time.

Common Ratio

In any geometric sequence, the common ratio is the factor you multiply to get from one term to the next. This ratio is a crucial component as it defines the behavior of the sequence. In our exercise, the sequence is expressed as \(a_{n} = 8(-0.75)^{n-1}\), with a common ratio of -0.75.

A negative common ratio like -0.75 means that the terms in the sequence alternate between positive and negative. As you multiply each previous term by a number less than 1 in absolute value, the size of each term decreases over time, leading the sequence values closer to zero. This type of geometric sequence is both shrinking and oscillating, showcasing a distinctive behavior on a graph.

Sequence Terms Calculation

Calculating the terms of a geometric sequence involves substituting different values of \(n\) into the given expression. For this sequence described by \(a_{n} = 8(-0.75)^{n-1}\), you begin with \(n = 1\). To find each sequential term, repeat the substitution process up to \(n = 10\).

Here's how it is structured:

• First term (\(n=1\)): \(a_{1} = 8(-0.75)^{0} = 8\)
• Second term (\(n=2\)): \(a_{2} = 8(-0.75)^{1} = -6\)
• Third term (\(n=3\)): \(a_{3} = 8(-0.75)^{2} = 4.5\)

Continue this process through to \(n=10\) to calculate all terms.The substitution helps you derive not just the 10 numbers needed to later input into a graphing utility, but also enhances understanding of how the sequence behaves over time. Each step reveals a term of the sequence, providing insight into the effect of the common ratio and contributing to the eventual graph.