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: Understanding the Formula

The nth term of the sequence is given by \(a_{n}=\frac{5}{2}\left(-\frac{1}{2}\right)^{n-1}\). This formula involves a geometric sequence, where each term can be calculated directly by substituting \(n\) into the formula.

Step 2: Calculating the First Term

To find the first term \(a_1\), substitute \(n = 1\) into the formula: \(a_{1}=\frac{5}{2}\left(-\frac{1}{2}\right)^{1-1} = \frac{5}{2}\left(-\frac{1}{2}\right)^{0} = \frac{5}{2}\cdot 1 = \frac{5}{2}\).

Step 3: Calculating the Second Term

To find the second term \(a_2\), substitute \(n = 2\) into the formula: \(a_{2}=\frac{5}{2}\left(-\frac{1}{2}\right)^{2-1} = \frac{5}{2}\left(-\frac{1}{2}\right) = -\frac{5}{4}\).

Step 4: Calculating the Third Term

To find the third term \(a_3\), substitute \(n = 3\) into the formula: \(a_{3}=\frac{5}{2}\left(-\frac{1}{2}\right)^{3-1} = \frac{5}{2}\left(\frac{1}{4}\right) = \frac{5}{8}\).

Step 5: Calculating the Fourth Term

To find the fourth term \(a_4\), substitute \(n = 4\) into the formula: \(a_{4}=\frac{5}{2}\left(-\frac{1}{2}\right)^{4-1} = \frac{5}{2}\left(-\frac{1}{8}\right) = -\frac{5}{16}\).

Step 6: Calculating the Fifth Term

To find the fifth term \(a_5\), substitute \(n = 5\) into the formula: \(a_{5}=\frac{5}{2}\left(-\frac{1}{2}\right)^{5-1} = \frac{5}{2}\left(\frac{1}{16}\right) = \frac{5}{32}\).

Step 7: Determine the Common Ratio

The sequence is geometric, with consecutive terms found by multiplying the previous term by a constant. The first term is \(\frac{5}{2}\), and the second term is \(-\frac{5}{4}\). The common ratio \(r\) is therefore the ratio of the second term to the first term: \(r = \frac{-\frac{5}{4}}{\frac{5}{2}} = -\frac{1}{2}\).

Step 8: Graph the First Five Terms

To graph the terms, plot the points \((1, \frac{5}{2})\), \((2, -\frac{5}{4})\), \((3, \frac{5}{8})\), \((4, -\frac{5}{16})\), and \((5, \frac{5}{32})\) on a coordinate plane. Connect these points to represent the sequence graphically.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio \(r\) is an essential element that defines the pattern of the sequence. It's a fixed number that you'll use to multiply each term to get to the next term.
For the sequence in the exercise, we found that the first term \((a_1)\) is \(\frac{5}{2}\), and the second term \((a_2)\) is \(-\frac{5}{4}\).
To find the common ratio, you need to divide the second term by the first term. Let's calculate: \[\frac{a_2}{a_1} = \frac{-\frac{5}{4}}{\frac{5}{2}} = -\frac{1}{2}\]This result tells us that the common ratio is \(-\frac{1}{2}\).
Notice how each term in the sequence is actually the previous term multiplied by this common ratio. This constant multiplication is what keeps the pattern of a geometric sequence consistent.

Exploring the n-th Term Formula

The n-th term formula for a geometric sequence helps you find any term in the sequence without calculating all the previous terms. It is commonly presented as \(a_n = a_1 r^{n-1}\).
Here, \(a_1\) is the first term of the sequence, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence.
For the specific sequence given in the exercise, the formula is: \[a_{n} = \frac{5}{2}\left(-\frac{1}{2}\right)^{n-1}\]In this formula:

• \(a_1 = \frac{5}{2}\)
• \(r = -\frac{1}{2}\)
• Each value of \(n\) specifies which term you're calculating

Understanding this formula means you can quickly find any term in the sequence, which is particularly powerful for sequences with many terms.
Also, using this formula can help confirm the consistency and correctness of your calculated values, ensuring the pattern remains true.

Graphing Geometric Sequences

Graphing sequences, especially geometric ones, is an excellent way to visualize the behavior and pattern of the sequence terms.
In this exercise, you've calculated the first five terms: \(\frac{5}{2}, -\frac{5}{4}, \frac{5}{8}, -\frac{5}{16}, \frac{5}{32}\).
To create a graph, plot these values on a coordinate plane, with the x-axis representing the term number (or position \(n\)) and the y-axis representing the term value \(a_n\).
The points you plot will be:

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

Connect these points with a smooth curve or lines to observe the pattern visually.
The graph will typically show oscillations because of the alternating sign due to the negative common ratio, displaying a distinctive wave-like pattern. Graphs help you understand how quickly values grow or shrink, and they provide a clearer picture of how the sequence behaves as \(n\) gets larger.