Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graphing the Terms of a Sequence Use a graphing utility to graph the first 10 terms of the sequence. $$a_{n}=10(-1.2)^{n-1}$$

Step-by-Step Solution

Answer

The sequence can be plotted on a scatter plot, and will show a geometric sequence pattern with a common ratio of -1.2.

Step 1: Understand the sequence

The given sequence is \(a_{n}=10(-1.2)^{n-1}\). It is helpful to note that this is a geometric sequence where the first term \(a_{1}\) is 10 and the common ratio is -1.2.

Step 2: Calculate the first ten terms

Start with n=1 and substitute it into the sequence to find the first term. Repeat for n=2, n=3, ..., n=10 to get the first 10 terms of the sequence.

Step 3: Plotting the sequence

Use a graphing utility to plot the sequence. The x-values will be from 1 to 10 (representing the 'n' values) and the corresponding y-values will be the 10 terms of the sequence calculated in the previous step. This will give you a scatter plot, as sequences are discrete, not continuous.

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term is generated by multiplying the previous term by a fixed, non-zero number called the common ratio. It is useful in various mathematical scenarios due to its predictable pattern of growth or shrinkage. In the given exercise, the sequence is represented by the formula \[ a_{n} = 10(-1.2)^{n-1} \].

• The first term \( a_1 \) is found by substituting \( n = 1 \) into the formula, resulting in \( a_1 = 10 \times (-1.2)^{0} = 10 \).
• Each subsequent term is calculated by multiplying the previous term by the common ratio, which in this case is \(-1.2\).

Recognizing a geometric sequence is important for analyzing how quickly or drastically numbers change from one term to the next. This quality makes them very handy when you are looking to solve problems involving exponential growth or decay.

Common Ratio

The common ratio in a geometric sequence is the constant factor between consecutive terms. It defines whether the sequence is increasing, decreasing, or oscillating. In our specific exercise, the common ratio is \(-1.2\).

• When the common ratio is positive, the terms tend to grow or shrink uniformly.
• If the common ratio is negative, as in this example, the terms not only change in size but also alternate in sign.

This alternation is evident in the sequence \( a_n = 10(-1.2)^{n-1} \), where the sequence oscillates due to the negative common ratio \(-1.2\). Such sequences can show behaviors like alternating between positive and negative values, which are important for understanding more dynamic or cyclical patterns in mathematics.

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical sequences and functions. These include both physical devices, like graphing calculators, and software applications available online. Using a graphing utility is especially helpful for sequences because it allows for a clear visual representation, making the relationships and behaviors within the sequence more apparent.When you input the sequence formula \( a_{n} = 10(-1.2)^{n-1} \) into a graphing utility:

• Set your x-values to start from 1 to 10, which represent the first ten terms (or 'n' values).
• The graphing utility then calculates and plots the y-values based on the given formula, providing a visual insight into how the sequence behaves.

Using this visual aid simplifies complex algebraic concepts by showing them in an intuitive, graphical form, thereby helping you better grasp the sequence's progression and behavior.

Scatter Plot

A scatter plot is a type of graph used to represent discrete data points, such as terms of a sequence. It is particularly common in statistics and mathematics to depict the relationship between two variables.In this exercise, the x-axis represents the term number \( n \), while the y-axis depicts the corresponding term value \( a_n \). Each pair of (n, a_n) is plotted as a distinct point:

• The plotted points for the sequence \( a_n = 10(-1.2)^{n-1} \) are scattered across the graph because sequences are not continuous.
• This spacing illustrates the oscillating nature of the terms due to the negative common ratio.

The scatter plot helps to visually highlight any rising, falling, or alternating patterns within the sequence. By viewing the plotted points, you can easily analyze and interpret the sequence's overall trend as well as identifying any unique behaviors it may exhibit.