Question

In Exercises 23-30, write a rule for the \(n\)th term. Then graph the first six terms of the sequence. $$ a_5=3, r=-\frac{1}{3} $$

Short answer

The rule is \(a_n = 243 \cdot (-1/3)^{n-1}\) and after plotting the first six terms, the graph bounces between positive and negative values, decreasing in magnitude.

Step-by-step solution

Determine the rule for the nth term

A geometric sequence follows the formula \(a_n=a_1 \cdot r^{n-1}\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio and \(n\) is the term number. Here, we are given \(a_5=3\) and \(r=-1/3\). We need to find out the value of \(a_1\). So, the formula for our exercise becomes \(3=a_1 \cdot (-1/3)^{5-1}\). Simplifying gives \(a_1 = 3 \cdot 3^4= 243\). Therefore the rule for the nth term is \(a_n = 243 \cdot (-1/3)^{n-1}\).

Compute the first six terms

Following the formula of the nth term, \(a_n = 243 \cdot (-1/3)^{n-1}\), calculate the first six terms (\(a_1\) to \(a_6\)).

Graph the sequence

After computing the first six terms, plot these points on a graph using the term number as x-coordinate and the term value as y-coordinate. Be sure to label the points properly and draw the connecting lines between the points.

Key concepts

nth Term Rule

Geometric sequences follow a special formula to define the general term for any position in the sequence. This formula is known as the rule for the nth term. The rule helps determine the value of any term in the sequence without calculating all the previous terms one by one.

To find the nth term in a geometric sequence, we use the expression \(a_n = a_1 \cdot r^{n-1}\). Here:

• \(a_n\) represents the nth term, the term we want to find.
• \(a_1\) stands for the first term of the sequence.
• \(r\) is the common ratio, which we'll discuss shortly.
• \(n\) is the term number, or which position in the sequence we're looking for.

In our exercise, we needed to calculate \(a_1\), since it was not directly provided. We used \(a_5 = 3\) and the given common ratio \(r = -\frac{1}{3}\) in the formula to work backward and find that \(a_1 = 243\). This allowed us to write the nth term rule for our specific sequence as \(a_n = 243 \cdot (-\frac{1}{3})^{n-1}\).

This formula is crucial as it allows us to generate terms of the sequence quickly and effortlessly, opening the doors to understanding other sequence behaviors and properties.

Common Ratio

In a geometric sequence, the common ratio is a vital component that defines how the sequence progresses. It is the consistent factor between consecutive terms.

To find the common ratio between terms, divide any term by its preceding term. In our exercise, the common ratio \(r = -\frac{1}{3}\) was given directly. This means each term in the sequence is \(-\frac{1}{3}\) times the previous term.

The common ratio can be positive or negative, leading to different sequence behaviors:

• If \(|r| < 1\), the terms will trend towards zero.
• If \(|r| = 1\), the sequence will be constant (not common in geometric sequences).
• If \(|r| > 1\), the terms will grow larger in magnitude.
• If \(r\) is negative, the terms will alternate in sign, causing an oscillating pattern as seen in our sequence.

Understanding the common ratio is key to predicting the sequence's behavior and applying it to real-world scenarios where geometric sequences are prevalent, such as interest rates or population growth models.

Graphing Sequences

Graphing sequences helps visualize the pattern and progression, making it easier to grasp the concept of a geometric sequence.

When graphing a geometric sequence, plot each term using its position in the sequence (the term number) as the x-coordinate and the term's value as the y-coordinate.

• Start by calculating and plotting the first few terms using their sequence formula, much like calculating points in other mathematical graphs.
• Ensure the points are clearly labeled to avoid confusion, especially when terms alternate using negative common ratios.
• Connect the points with a line to show the pattern, although a geometric sequence is technically a series of discrete points.

In our specific case, after computing the first six terms using the nth term formula, graphing them reveals the alternating sign nature due to the negative common ratio. Such visualizations not only aid in comprehension but also in identifying trends and verifying calculations.