Problem 48 - College Algebra

Question

For the following exercises, use the information provided to graph the first five terms of the geometric sequence. $$ a_{1}=1, r=\frac{1}{2} $$

Step-by-Step Solution

Answer

The first five terms are: 1, 0.5, 0.25, 0.125, and 0.0625.

Step 1: Identify the Formula for Geometric Sequence

The formula to find the terms of a geometric sequence is given by: \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 2: Calculate the First Term

The first term of the sequence is given as \( a_1 = 1 \). Thus, the first term is 1.

Step 3: Calculate the Second Term

To find the second term, use the formula with \( n=2 \): \( a_2 = 1 \cdot \left( \frac{1}{2} \right)^{(2-1)} = \frac{1}{2} \).

Step 4: Calculate the Third Term

For the third term, substitute \( n=3 \) into the formula: \( a_3 = 1 \cdot \left( \frac{1}{2} \right)^{(3-1)} = \frac{1}{4} \).

Step 5: Calculate the Fourth Term

The fourth term is obtained by setting \( n=4 \): \( a_4 = 1 \cdot \left( \frac{1}{2} \right)^{(4-1)} = \frac{1}{8} \).

Step 6: Calculate the Fifth Term

Compute the fifth term using \( n=5 \): \( a_5 = 1 \cdot \left( \frac{1}{2} \right)^{(5-1)} = \frac{1}{16} \).

Step 7: Plot the Terms on the Graph

On graph paper or a digital tool, plot the points corresponding to the terms: \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \). Place these points on the y-axis against their respective position numbers on the x-axis (1 through 5).

Key Concepts

Graphing sequences

Graphing a sequence can give you a visual representation of how the terms relate to each other. In the case of a geometric sequence, the points on the graph will typically show an exponential trend. Choose the x-axis to represent the term number and the y-axis to represent the value of the term.
Some steps to graph a geometric sequence:

• Identify the first few terms using the formula, if not given.
• Label the x-axis with term numbers starting from 1.
• Label the y-axis with values appropriate for your terms.
• Plot each term as a point, connecting them lightly as desired to reflect the exponential nature.

For this sequence, plot the points (1, 1), (2, 0.5), (3, 0.25), (4, 0.125), and (5, 0.0625) on your graph.
This will depict how the sequence diminishes exponentially. Each plotted point is typically a fraction of the previous value, creating a downward slope that gets shallower as it progresses.

Common ratio

In a geometric sequence, the common ratio is the factor by which we multiply one term to get the next term. It remains consistent throughout the sequence, making it a key part of defining the sequence's behavior.
Some features of a common ratio:

• If the common ratio is greater than 1, the sequence will expand positively.
• A ratio between 0 and 1 leads to a sequence that decreases, approaching zero but never quite hitting it.
• If the ratio is negative, terms will alternate in sign, leading to a diverging oscillating pattern.

In the exercise example, the common ratio is \( \frac{1}{2} \). This indicates that each term is half of the previous term, causing the sequence to shrink progressively.
Understanding the common ratio helps pinpoint the general trend of the sequence, whether it's shrinking, growing, or oscillating.

Mathematical sequences

Mathematical sequences are ordered lists of numbers that follow a particular rule or pattern. In sequences, identifying the type can aid in determining their properties and applications. A geometric sequence, like the one in the exercise, multiplies each term by a consistent value.
Important characteristics of sequences:

• A sequence can be infinite, having no end, or finite.
• The type of sequence dictates the formula or method used to find terms.
• Types of sequences include arithmetic, geometric, and others defined by unique rules.

Understanding sequences means recognizing the importance of their rules in predicting future terms and solving problems.
In this exercise, using the formula \( a_n = a_1 \cdot r^{(n-1)} \), you find the sequence terms efficiently, indicating the sequence's trend and behavior over time.