Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, graph the first five terms of the indicated sequence \(a_{n}=\frac{(-1)^{n}}{n}+n\)

Step-by-Step Solution

Answer

The first five terms of the sequence are 0, 2.5, 2.667, 4.25, and 4.8.

Step 1: Understand the Sequence

The sequence given is \(a_{n} = \frac{(-1)^{n}}{n} + n\). This function includes both an alternating component \(\frac{(-1)^{n}}{n}\) and a linear component \(n\). The alternating component makes the terms switch signs depending on whether \(n\) is odd or even.

Step 2: Calculate the First Five Terms

Substitute \(n = 1, 2, 3, 4, 5\) into the sequence formula:- For \(n = 1\): \(a_1 = \frac{(-1)^{1}}{1} + 1 = -1 + 1 = 0\)- For \(n = 2\): \(a_2 = \frac{(-1)^{2}}{2} + 2 = \frac{1}{2} + 2 = 2.5\)- For \(n = 3\): \(a_3 = \frac{(-1)^{3}}{3} + 3 = -\frac{1}{3} + 3 = 2.667\)- For \(n = 4\): \(a_4 = \frac{(-1)^{4}}{4} + 4 = \frac{1}{4} + 4 = 4.25\)- For \(n = 5\): \(a_5 = \frac{(-1)^{5}}{5} + 5 = -\frac{1}{5} + 5 = 4.8\)

Step 3: Plot the Points on a Graph

Plot the points (1, 0), (2, 2.5), (3, 2.667), (4, 4.25), and (5, 4.8) on a coordinate grid. These points represent the terms of the sequence where the x-axis corresponds to the term number \(n\) and the y-axis corresponds to the value of the sequence \(a_{n}\). Connect these points to see how the sequence behaves.

Key Concepts

Alternating Series

An alternating series is a sequence of numbers where the sign of the terms switches between positive and negative. This flipping of signs is visually recognized when you plot the terms on a graph and observe them jumping from below the x-axis to above, or vice-versa. In our sequence, \(a_n = \frac{(-1)^n}{n} + n\), the alternating portion is represented by \(\frac{(-1)^n}{n}\).

• For an odd \(n\), \((-1)^n\) equals \(-1\), making the term negative.
• For an even \(n\), \((-1)^n\) equals \(1\), making the term positive.

Here, even values of \(n\) will push terms above their linear path \(n\), while odd values will nudge them below. This zigzagging behavior helps form the alternating pattern we see.

Term Calculation

Calculating the terms of a sequence involves plugging in successive integers for \(n\). This is straightforward but requires careful attention to the signs, especially in alternating series. For instance, calculating the first five terms of \(a_n = \frac{(-1)^n}{n} + n\) yields:

• \(n = 1:\) \(a_1 = 0\), because the negative part cancels out the linear part.
• \(n = 2:\) \(a_2 = 2.5\), as the positive alteration adds to the linear portion.
• \(n = 3:\) \(a_3 = 2.667\), where the term slightly drops below due to the negative factor.
• \(n = 4:\) \(a_4 = 4.25\), the positive term is once again boosted up.
• \(n = 5:\) \(a_5 = 4.8\), the sequence dips slightly downwards as the negative influence reappears.

This cycle reveals how alternating sequences retain a regular and predictable pattern when terms are computed.

Coordinate Plotting

Coordinate plotting translates the terms of a sequence into points on a grid. Each term \(a_n\) is assigned a coordinate \((n, a_n)\), where \(n\) is the term number (plotted on the x-axis) and \(a_n\) is its value (plotted on the y-axis). Here's how to approach plotting:

• Start by setting up your coordinate plane with axes clearly labeled.
• Mark points: for this sequence use (1, 0), (2, 2.5), (3, 2.667), (4, 4.25), and (5, 4.8).
• Connect these points to visualize the trend or pattern of the sequence.

Each plotted point joins the previous to illustrate how the sequence behaves, allowing you to see patterns, like the effect of alternating terms and linear growth. This makes abstract sequences concrete, facilitating a deeper understanding of their behavior.