Problem 7 - Precalculus Mathematics for Calculus

Question

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$a_{n}=\frac{5}{2}\left(-\frac{1}{2}\right)^{n-1}$$

Step-by-Step Solution

Answer

First five terms: \(\frac{5}{2}, -\frac{5}{4}, \frac{5}{8}, -\frac{5}{16}, \frac{5}{32}\); common ratio \(r = -\frac{1}{2}\).

Step 1: Identify the Formula

The formula for the nth term of the sequence is given by \(a_{n}=\frac{5}{2}\left(-\frac{1}{2}\right)^{n-1}\). This means each term in the sequence is derived using this expression, where \(n\) is the position of the term in the sequence.

Step 2: Calculate the First Five Terms

To find the first five terms, substitute \(n = 1, 2, 3, 4, 5\) into the formula: \(a_{1} = \frac{5}{2}\left(-\frac{1}{2}\right)^{0} = \frac{5}{2}\), \(a_{2} = \frac{5}{2}\left(-\frac{1}{2}\right)^{1} = -\frac{5}{4}\), \(a_{3} = \frac{5}{2}\left(-\frac{1}{2}\right)^{2} = \frac{5}{8}\), \(a_{4} = \frac{5}{2}\left(-\frac{1}{2}\right)^{3} = -\frac{5}{16}\), \(a_{5} = \frac{5}{2}\left(-\frac{1}{2}\right)^{4} = \frac{5}{32}\). Thus, the first five terms are: \(\frac{5}{2}, -\frac{5}{4}, \frac{5}{8}, -\frac{5}{16}, \frac{5}{32}\).

Step 3: Determine the Common Ratio

To find the common ratio \(r\), observe that the given sequence is geometric because the formula involves a power of a constant \(-\frac{1}{2}\). This constant is the common ratio \(r = -\frac{1}{2}\).

Step 4: Plot the Graph of the Terms

Plot the first five terms on a graph where the x-axis represents the term number \(n\) and the y-axis represents the term value \(a_{n}\). The points to plot are: \((1, \frac{5}{2}), (2, -\frac{5}{4}), (3, \frac{5}{8}), (4, -\frac{5}{16}), (5, \frac{5}{32})\). Connect these points to visualize the sequence.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is the key feature that transforms one term into the next. It is the constant factor by which each term is multiplied to yield the subsequent term. This can make these sequences increase or decrease quickly depending on the value of the ratio.

In the example given, the common ratio is easily identified within the term formula: \(-\frac{1}{2}\). Each term results from multiplying the previous term by this ratio:

• The second term \(-\frac{5}{4}\) is obtained by multiplying the first term \(\frac{5}{2}\) with the common ratio \(-\frac{1}{2}\).
• Continuing this pattern confirms the consistency of the geometric sequence.

The common ratio plays a crucial role in determining the direction and nature of the sequence. Here, the ratio is negative, resulting in terms that alternate in sign, producing an oscillating pattern.

The Process of Term Calculation

To calculate any term in a geometric sequence, we rely heavily on its standard formula. For the sequence given by \(a_{n}=\frac{5}{2}\left(-\frac{1}{2}\right)^{n-1}\), each term is determined by specifying the corresponding \(n\).

Calculating involves simplifying the expression for each selected term number:

• Substitute \(n=1\) to find the first term resulting in \(\frac{5}{2}\).
• For the second term, set \(n=2\) in the formula, simplifying to yield \(-\frac{5}{4}\).
• Continue this substitution process for the third, fourth, and fifth terms leading to \(\frac{5}{8}, -\frac{5}{16},\) and \(\frac{5}{32}\) respectively.

This method showcases how easily each term can be calculated once a formula is known, providing crucial insights into the sequence's characteristics.

Graphing Geometric Sequences Effectively

Graphing the terms of a geometric sequence gives a visual illustration of its behavior. Each term is plotted as a point on a graph where the x-axis signifies the term number and the y-axis denotes the term's value.

This is done by organizing each term-pair like so:

• The first point plotted is \((1, \frac{5}{2})\).
• Then follows \((2, -\frac{5}{4})\), moving on to \((3, \frac{5}{8})\).
• Graph \((4, -\frac{5}{16})\) and finally \((5, \frac{5}{32})\).

Upon graphing, traces of connection often emerge depicting the progression visually. A distinct characteristic of alternating sequences caused by a negative ratio mirrors through points lying above and below the x-axis alternately. This gives learners a tangible perspective on how geometric sequences unfold similarly with real-life data.