Problem 7 - 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

The first five terms are \(\frac{5}{2}, -\frac{5}{4}, \frac{5}{8}, -\frac{5}{16}, \frac{5}{32}\). The common ratio is \(-\frac{1}{2}\). Graph the points on an \(n\) versus \(a_n\) chart.

Step 1: Find the First Term

To find the first term of the sequence, substitute \(n = 1\) into the expression for \(a_n\): \[ a_1 = \frac{5}{2} \left(-\frac{1}{2}\right)^{1-1} = \frac{5}{2} \times 1 = \frac{5}{2}. \] So, the first term is \(\frac{5}{2}\).

Step 2: Find the Second Term

Substitute \(n = 2\) into the expression for \(a_n\): \[ a_2 = \frac{5}{2} \left(-\frac{1}{2}\right)^{2-1} = \frac{5}{2} \times \left( -\frac{1}{2} \right) = -\frac{5}{4}. \] Thus, the second term is \(-\frac{5}{4}\).

Step 3: Find the Third Term

Substitute \(n = 3\) into the expression for \(a_n\): \[ a_3 = \frac{5}{2} \left(-\frac{1}{2}\right)^{3-1} = \frac{5}{2} \times \frac{1}{4} = \frac{5}{8}. \] So, the third term is \(\frac{5}{8}\).

Step 4: Find the Fourth Term

Substitute \(n = 4\) into the expression for \(a_n\): \[ a_4 = \frac{5}{2} \left(-\frac{1}{2}\right)^{4-1} = \frac{5}{2} \times \left(-\frac{1}{8}\right) = -\frac{5}{16}. \] Thus, the fourth term is \(-\frac{5}{16}\).

Step 5: Find the Fifth Term

Substitute \(n = 5\) into the expression for \(a_n\): \[ a_5 = \frac{5}{2} \left(-\frac{1}{2}\right)^{5-1} = \frac{5}{2} \times \frac{1}{16} = \frac{5}{32}. \] So, the fifth term is \(\frac{5}{32}\).

Step 6: Determine the Common Ratio

The common ratio \(r\) of a geometric sequence is found by dividing a term by its preceding term. Using the first two terms: \[ r = \frac{a_2}{a_1} = \frac{-\frac{5}{4}}{\frac{5}{2}} = -\frac{1}{2}. \] Hence, the common ratio \(r\) is \(-\frac{1}{2}\).

Step 7: Graph the Sequence Terms

Plot the calculated terms on a graph with the x-axis representing the term position \(n\) and the y-axis representing the sequence values. Points to be plotted 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 geometric progression.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is key to understanding how the sequence evolves. It's the factor by which each term is multiplied to get the next term.
For our given sequence: \[ a_{n} = \frac{5}{2}\left(-\frac{1}{2}\right)^{n-1} \] To find the common ratio \(r\), you need to take any term and divide it by the previous term. Here's how this looks:

• Start with the first term \(a_1 = \frac{5}{2}\) and the second term \(a_2 = -\frac{5}{4}\).
• Calculate the ratio: \[ r = \frac{a_2}{a_1} = \frac{-\frac{5}{4}}{\frac{5}{2}} = -\frac{1}{2}. \]

Thus, the common ratio here is \(-\frac{1}{2}\). Understanding this ratio helps reveal the underlying pattern of the sequence. In this case, each term is multiplied by \(-\frac{1}{2}\) to produce the next term. This alternating sign and decreasing magnitude is a hallmark of a geometric sequence with a negative common ratio.

Exploring Sequence Terms

Sequence terms are the individual elements that make up a sequence. For any sequence described by a formula, you can find specific terms by plugging integers into the formula.
Let's find the first five terms of our sequence:

• First term: plug in \(n = 1\)
  1. \( a_1 = \frac{5}{2}(1) = \frac{5}{2} \)
  2. Hence, \(a_1 = \frac{5}{2} \)
• Second term: plug in \(n = 2\)
  1. \( a_2 = \frac{5}{2}\left(-\frac{1}{2}\right)^1 = -\frac{5}{4} \)
  2. Hence, \(a_2 = -\frac{5}{4} \)
• Third term: plug in \(n = 3\)
  1. \( a_3 = \frac{5}{2}\left(-\frac{1}{2}\right)^2 = \frac{5}{8} \)
  2. Hence, \(a_3 = \frac{5}{8} \)
• Fourth term: plug in \(n = 4\)
  1. \( a_4 = \frac{5}{2}\left(-\frac{1}{2}\right)^3 = -\frac{5}{16} \)
  2. Hence, \(a_4 = -\frac{5}{16} \)
• Fifth term: plug in \(n = 5\)
  1. \( a_5 = \frac{5}{2}\left(-\frac{1}{2}\right)^4 = \frac{5}{32} \)
  2. Hence, \(a_5 = \frac{5}{32} \)

By systematically finding these terms, you can see the pattern emerging due to the repeated multiplication by the common ratio. This kind of structured handling of sequence terms is foundational in exploring sequences mathematically.

Graphing Sequences for Visual Understanding

Graphing sequences provides a visual insight into the behavior and characteristics of sequences that are sometimes difficult to perceive numerically.
To graph the sequence terms we've calculated above, set up a coordinate system where:

• the x-axis represents the term position \(n\)
• the y-axis represents the value of the sequence terms \(a_n\)

Let’s plot the following calculated points:

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

After plotting these points, connect them sequentially to visualize the trend of the sequence.
Notice how the points alternate in sign and decrease in magnitude. This reflects the nature of the common ratio we've identified: \(-\frac{1}{2}\).
Graphing allows for an intuitive understanding, revealing how the geometric sequence progresses and how each term compares to the rest. This method enhances comprehension by enabling learners to see mathematical relationships in action beyond just numerical or algebraic expressions.