Problem 3 - Precalculus: Mathematical 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

(a) Terms: \(\frac{5}{2}, -\frac{5}{4}, \frac{5}{8}, -\frac{5}{16}, \frac{5}{32}\). (b) Ratio \(r\): \(-\frac{1}{2}\). (c) Plot terms on graph.

Step 1: Write the Formula for the n-th Term

The n-th term of the sequence is given by the formula \(a_n = \frac{5}{2} \left(-\frac{1}{2}\right)^{n-1}\). This formula will be used to find the specific terms by plugging in values of \(n\).

Step 2: Calculate the First Five Terms

To find the first five terms, we'll substitute \(n = 1, 2, 3, 4,\) and \(5\) into the n-th term formula.- For \(n = 1\): \(a_1 = \frac{5}{2} \left(-\frac{1}{2}\right)^{1-1} = \frac{5}{2} \cdot 1 = \frac{5}{2}\).- For \(n = 2\): \(a_2 = \frac{5}{2} \left(-\frac{1}{2}\right)^{2-1} = \frac{5}{2} \left(-\frac{1}{2}\right) = -\frac{5}{4}\).- For \(n = 3\): \(a_3 = \frac{5}{2} \left(-\frac{1}{2}\right)^{3-1} = \frac{5}{2} \left(\frac{1}{4}\right) = \frac{5}{8}\).- For \(n = 4\): \(a_4 = \frac{5}{2} \left(-\frac{1}{2}\right)^{4-1} = \frac{5}{2} \left(-\frac{1}{8}\right) = -\frac{5}{16}\).- For \(n = 5\): \(a_5 = \frac{5}{2} \left(-\frac{1}{2}\right)^{5-1} = \frac{5}{2} \left(\frac{1}{16}\right) = \frac{5}{32}\).So 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

Since this is a geometric sequence, the common ratio \(r\) is the factor multiplied to get from one term to the next. Observe the consecutive terms:- From \(\frac{5}{2}\) to \(-\frac{5}{4}\): \(-\frac{5}{4} \div \frac{5}{2} = -\frac{1}{2}\).- From \(-\frac{5}{4}\) to \(\frac{5}{8}\): \(\frac{5}{8} \div -\frac{5}{4} = -\frac{1}{2}\).The common ratio is confirmed as \(-\frac{1}{2}\).

Step 4: Graph the First Five Terms

Plot each of the first five terms as points on a graph where the x-axis represents the term number (1 to 5) and the y-axis represents the value of the terms. The points will be: - (1, \(\frac{5}{2}\))- (2, \(-\frac{5}{4}\))- (3, \(\frac{5}{8}\))- (4, \(-\frac{5}{16}\))- (5, \(\frac{5}{32}\))These points should create a zig-zag pattern demonstrating the alternating nature of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the fixed number that you multiply by each term to get to the next term. Understanding the common ratio is crucial as it defines how the sequence progresses.

In our example exercise, we find each term of the sequence by using the formula given for the n-th term. To spot the common ratio, we take two consecutive terms and divide the latter by the former. Let's detail this process a bit.
- Take two consecutive terms: such as \(a_1 = \frac{5}{2}\) and \(a_2 = -\frac{5}{4}\)

• Then divide \(a_2\) by \(a_1: -\frac{5}{4} \div \frac{5}{2} = -\frac{1}{2}\)

Thus, the common ratio \(r\) for this sequence is \(-\frac{1}{2}\).
This negative ratio means that the signs of the sequence terms will alternate, with each term getting progressively closer to a limit.

Sequence Terms

Sequence terms are the elements or numbers in a sequence, listed in a specific order based on a given formula. To determine these terms, you substitute successive natural numbers into the formula.

For our geometric sequence in practice, we use the formula provided: \a_n = \frac{5}{2} \left(-\frac{1}{2}\right)^{n-1}\.
Here, we calculate the first five terms by putting in values \(n=1, 2, 3, 4, 5\) into the formula.

• When \(n=1\), \(a_1 = \frac{5}{2} \)
• When \(n=2\), \(a_2 = -\frac{5}{4}\)
• When \(n=3\), \(a_3 = \frac{5}{8}\)
• When \(n=4\), \(a_4 = -\frac{5}{16}\)
• When \(n=5\), \(a_5 = \frac{5}{32}\)

These terms follow a particular pattern defined by their original formula and are reflective of the behavior dictated by the common ratio.

Graphing Sequences

Graphing sequences involves plotting the terms onto a graph to visualize their progression and to identify patterns.

In our exercise, we take the sequence numbers we have calculated for \(n=1, 2, 3, 4, 5\) and plot them on a coordinate graph.
- The x-axis represents each term number (i.e., 1, 2, 3, etc).
- The y-axis represents the value of each corresponding term, such as \(a_1 = \frac{5}{2}\), \(a_2 = -\frac{5}{4}\), etc.

• Plot (1, \(\frac{5}{2}\))
• Plot (2, \(-\frac{5}{4}\))
• Plot (3, \(\frac{5}{8}\))
• Plot (4, \(-\frac{5}{16}\))
• Plot (5, \(\frac{5}{32}\))

Connecting these plotted points showcases a zig-zag pattern formed by the sequence, highlighting its alternating nature due to the negative common ratio. Graphing offers a visual insight into the behavior and progression of sequence terms.