Problem 81 - 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.) $$a_{n}=\frac{2}{3} n$$

Step-by-Step Solution

Answer

The sequence \(a_{n}=\frac{2}{3} n\) for the first 10 terms is furnished by the points (1, \(\frac{2}{3}\)), (2, \(\frac{4}{3}\)), ..., (10, \(\frac{20}{3}\)). Using a graphing utility, plot these coordinates to visualize the sequence.

Step 1: Identify sequence

The sequence given in the problem is \(a_{n}=\frac{2}{3} n\), where \(n\) represents the term number in the sequence. This formula tells us that to find the value of any term in the sequence, we multiply the term number by \(\frac{2}{3}\).

Step 2: Generate the sequence

Let's generate the first 10 terms of the sequence by substituting the values of \(n\) from 1 to 10 into the given expression. The sequence will look like this: \(a_1\) = \(\frac{2}{3}\) * 1, \(a_2\) = \(\frac{2}{3}\) * 2, ..., \(a_{10}\) = \(\frac{2}{3}\) * 10.

Step 3: Calculate the terms of the sequence

Now we calculate the values of the sequence we generated in the previous step. This results in: \(a_1 = \frac{2}{3}, a_2 = \frac{4}{3}, ..., a_{10} = \frac{20}{3}\).

Step 4: Plot the sequence

The resulting sequence forms a simple linear sequence where the points are: (1, \(a_1\)), (2, \(a_2\)), ..., (10, \(a_{10}\)). To plot these on a graph, place dots at the given coordinates and connect them to visualize the sequence.

Key Concepts

Graphing Sequences

Graphing sequences can help us visualize and better understand how a sequence behaves as it progresses. In our example, we first need to calculate a sequence's terms, then plot these terms against the term numbers.

• Each term in the sequence is calculated from a formula, in this case, \(a_{n} = \frac{2}{3}n\).
• For instance, when \(n = 1\), we find \(a_1 = \frac{2}{3}\), when \(n = 2\), \(a_2 = \frac{4}{3}\), and so on up to \(n = 10\).
• These terms are plotted as points on a graph, where the x-axis represents the term number \(n\), and the y-axis represents the value of \(a_n\).

By connecting these points, we can see the pattern the sequence forms. In this case, you will notice a straight line, indicating the regular increase in the sequence values. Graphs like this make it easy to quickly spot trends or patterns in sequences.

Linear Sequence

A linear sequence is a type of progression where the difference between consecutive terms is constant. This is often found in sequences defined by a linear formula, such as \(a_{n} = \frac{2}{3}n\). Here, each term increases by a fixed amount compared to the previous term.

• In a linear sequence, the relationship between the terms is straightforward and predictable, making calculations easy.
• The constant increase is determined by the coefficient of \(n\) in the sequence's formula. In this example, the sequence increases by \(\frac{2}{3}\) for each increase of 1 in \(n\).
• The plot of a linear sequence will always form a straight line, reflecting its consistent rate of change.

Understanding linear sequences helps in distinguishing them from other complex types of sequences, such as quadratic or exponential sequences, which do not have a constant rate of change.

Term Calculation

Calculating the terms of a sequence involves using a given formula to determine each term's value based on its position. The position is usually represented by \(n\), the term number.

• To calculate a term, substitute the position number \(n\) into the sequence's formula. For this sequence, \(a_{n} = \frac{2}{3}n\), the process involves simple multiplication.
• The first term \(a_1\), when \(n = 1\), is \(\frac{2}{3} \times 1 = \frac{2}{3}\).
• Continuing this calculation for each number up to \(n = 10\) yields the complete sequence: \(a_1 = \frac{2}{3}, a_2 = \frac{4}{3}, \, \ldots \, , a_{10} = \frac{20}{3}\).

This straightforward calculation reveals each term's systematic progression, which is easy to follow due to the linear nature of the sequence. Practicing term calculations is essential for mastering the prediction and analysis of sequence patterns.