Problem 4 - Algebra and Trigonometry

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 \( r = -\frac{1}{2} \). Graph the points accordingly.

Step 1: Substitute Values for First Five Terms

To find the first five terms of the sequence, substitute the values of \( n \) from 1 to 5 into the formula \( a_n = \frac{5}{2}\left(-\frac{1}{2}\right)^{n-1} \).

Step 2: Calculate Each Term

- For \( n=1 \), \( a_1 = \frac{5}{2}\left(-\frac{1}{2}\right)^{0} = \frac{5}{2}\cdot 1 = \frac{5}{2} \).- For \( n=2 \), \( a_2 = \frac{5}{2}\left(-\frac{1}{2}\right)^{1} = \frac{5}{2} \cdot -\frac{1}{2} = -\frac{5}{4} \).- For \( n=3 \), \( a_3 = \frac{5}{2}\left(-\frac{1}{2}\right)^{2} = \frac{5}{2} \cdot \frac{1}{4} = \frac{5}{8} \).- For \( n=4 \), \( a_4 = \frac{5}{2}\left(-\frac{1}{2}\right)^{3} = \frac{5}{2} \cdot -\frac{1}{8} = -\frac{5}{16} \).- For \( n=5 \), \( a_5 = \frac{5}{2}\left(-\frac{1}{2}\right)^{4} = \frac{5}{2} \cdot \frac{1}{16} = \frac{5}{32} \).

Step 3: List the First Five Terms

The first five terms of the sequence are: \( \frac{5}{2}, -\frac{5}{4}, \frac{5}{8}, -\frac{5}{16}, \frac{5}{32} \).

Step 4: Identify the Common Ratio

To identify the common ratio \( r \) of the geometric sequence, divide the second term by the first term, \( r = \frac{a_2}{a_1} = \frac{-\frac{5}{4}}{\frac{5}{2}} = -\frac{1}{2} \).

Step 5: Graph the Terms

Plot the terms on a graph where the x-axis represents the term number \( n \) and the y-axis represents the term value. 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 the points to observe the pattern.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the factor by which we multiply each term to get the next term. This ratio is a constant and plays a crucial role in determining the nature and behavior of the sequence.
To find the common ratio, divide any term in the sequence by the preceding term. For example, given the first two terms of our sequence, \( a_1 = \frac{5}{2} \) and \( a_2 = -\frac{5}{4} \), we can calculate the common ratio \( r \) as follows:

• \( r = \frac{a_2}{a_1} = \frac{-\frac{5}{4}}{\frac{5}{2}} = -\frac{1}{2} \)

This means each term in the sequence is \(-\frac{1}{2}\) times the previous term. A negative ratio, like \(-\frac{1}{2}\), also causes the terms to alternate between positive and negative.

Graphing Sequences

Graphing a sequence can help us visualize how the terms change and how they relate to each other. In this sequence, the graph will illustrate the alternating pattern due to the negative common ratio.
To create a graph for our sequence, follow these steps:

• Plot the first term \((1, \frac{5}{2})\).
• Move to the next term and plot \((2, -\frac{5}{4})\).
• Continue plotting the terms as \((3, \frac{5}{8}), (4, -\frac{5}{16}), \) and \((5, \frac{5}{32})\).

By connecting these points, most learners will identify a clear pattern emerging. The alternating signs in each term will reflect on the graph, shown by the points situated on either side of the x-axis.

Sequence Terms

Sequence terms refer to the individual numbers or expressions that constitute a sequence. In the context of a geometric sequence, they follow a pattern: each term is the product of the previous term and the common ratio.
To calculate the sequence terms, use the general formula for the nth term in a geometric sequence:

• \( a_n = a_1 \cdot r^{(n-1)} \)

In our example, the first term is \( a_1 = \frac{5}{2} \) and the common ratio \( r = -\frac{1}{2} \). Hence, for the first five terms,

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

Each step reaffirms the pattern of terms alternating in sign, following a simple multiplication by the negative common ratio.