Problem 39 - Precalculus with Limits

Question

In Exercises 37-42, use a graphing utility to graph the first 10 terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = 16 (-0.5)^{n - 1} \)

Step-by-Step Solution

Answer

The first 10 terms of the sequence \( a_n = 16 (-0.5)^{n - 1} \) are: 16, -8, 4, -2, 1, -0.5, 0.25, -0.125, 0.0625, -0.03125.

Step 1: Substituting the values

Plug the values of n from 1 to 10 into \( a_n = 16 (-0.5)^{n - 1} \). This will yield the first 10 terms of the sequence.

Step 2: Calculate the sequence

The calculated sequence for the first 10 terms is shown below: \n\( a_1 = 16 \times (-0.5)^{1 - 1} = 16 \)\( a_2 = 16 \times (-0.5)^{2 - 1}= -8 \)\( a_3 = 16 \times (-0.5)^{3 - 1} = 4 \)\( a_4 = 16 \times (-0.5)^{4 - 1} = -2 \)\( a_5 = 16 \times (-0.5)^{5 - 1} = 1 \)\( a_6 = 16 \times (-0.5)^{6 - 1} = -0.5 \)\( a_7 = 16 \times (-0.5)^{7 - 1} = 0.25 \)\( a_8 = 16 \times (-0.5)^{8 - 1} = -0.125 \)\( a_9 = 16 \times (-0.5)^{9 - 1} = 0.0625 \)\( a_{10} = 16 \times (-0.5)^{10 - 1} = -0.03125 \)

Step 3: Plotting the sequence on graph

Now plot the values onto a graph using any graphing tool. Lay out the x-values (terms) along the horizontal axis from 1 to 10, and the y-values (calculated values) vertically. The representation will be in form of dismissed points, as sequences are lists of numbers that are disconnected.

Key Concepts

Graphing Sequences

Graphing sequences is a visual way to understand the behavior and pattern of sequences. For geometric sequences like the one given in the exercise, observing the graph can help you see how the sequence alternates and decreases in size.

What makes graphing sequences unique is that unlike typical line graphs, sequence graphs appear as separate points. This is because sequences are discrete entities, representing specific, individual terms rather than a continuous flow.

By plotting these points, you get snapshots of different stages in the sequence. This visualization not only helps in analyzing the trend but also emphasizes any oscillation or pattern emerging from the defined terms.

First 10 Terms

To start graphing the sequence, calculating the first 10 terms is crucial. These calculations help us observe the sequence's initial direction and changes.

For the given geometric sequence:

• The formula is: \( a_n = 16 (-0.5)^{n - 1} \).
• Start by substituting \( n \) from 1 to 10.
• Note how the terms alternate between positive and negative, showcasing the nature of geometric sequences.

These first 10 terms are:

• \( a_1 = 16 \)
• \( a_2 = -8 \)
• \( a_3 = 4 \)
• \( a_4 = -2 \)
• \( a_5 = 1 \)
• \( a_6 = -0.5 \)
• \( a_7 = 0.25 \)
• \( a_8 = -0.125 \)
• \( a_9 = 0.0625 \)
• \( a_{10} = -0.03125 \)

Plotting Sequences

Plotting sequences requires placing the calculated terms onto a graph. Here's how you can do it:

• Use the horizontal axis to represent the term numbers (1 to 10).
• The vertical axis will show the sequence values you calculated.

Each point \((n, a_n)\) on the graph corresponds to a term of the sequence.

Since the sequence we're dealing with alternates negatives and positives, the points will hop above and below the x-axis. This zigzag pattern visually reinforces the concept of alternation in geometric sequences, especially when the common ratio (\(-0.5\) in this case) is negative.

Make sure each point is clearly marked to see the trend clearly. Graphing by hand or using a graphing calculator or utility tool will also provide insights into the spacing and behavior of those values.

Step by Step Solution

Understanding the step-by-step breakdown is key to mastering the graphing of sequences. Let's recap the process to ensure clarity:

• Step 1: **Calculate the Terms** - By substituting \( n = 1 \) to \( n = 10 \) into \( a_n = 16 (-0.5)^{n - 1} \), you determine each individual term. Remember, substitution is crucial to retrieving exact values.
• Step 2: **Check Sequence** - Ensure the calculated sequence values align with expected behaviors like alternating signs or reducing the absolute magnitudes, typical in geometric sequences.
• Step 3: **Graph the Sequence** - Finally, use your plotted points to visually confirm these patterns. Accounts for any potential arithmetic mistakes by comparing plotted dynamics with calculated values.

With this methodical approach, each step leads cleanly into the next, ensuring you not only find the correct terms but also understand the overarching pattern.