Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, use the information provided to graph the first five terms of the geometric sequence. \(a_{n}=27 \cdot 0.3^{n-1}\)

Step-by-Step Solution

Answer

Plot the points: (1, 27), (2, 8.1), (3, 2.43), (4, 0.729), (5, 0.2187).

Step 1: Identify the general formula

The formula given for the geometric sequence is \(a_{n} = 27 \cdot 0.3^{n-1}\). Here, \(27\) is the first term \(a_{1}\) when \(n\) is 1, and the common ratio \(r\) is \(0.3\).

Step 2: Calculate the first term

Substitute \(n = 1\) into the formula: \[ a_{1} = 27 \cdot 0.3^{1-1} = 27 \cdot 0.3^0 = 27 \cdot 1 = 27 \]. So, the first term is \(27\).

Step 3: Calculate the second term

Substitute \(n = 2\) into the formula: \[ a_{2} = 27 \cdot 0.3^{2-1} = 27 \cdot 0.3^1 = 27 \cdot 0.3 = 8.1 \]. So, the second term is \(8.1\).

Step 4: Calculate the third term

Substitute \(n = 3\) into the formula: \[ a_{3} = 27 \cdot 0.3^{3-1} = 27 \cdot 0.3^2 = 27 \cdot 0.09 = 2.43 \]. So, the third term is \(2.43\).

Step 5: Calculate the fourth term

Substitute \(n = 4\) into the formula: \[ a_{4} = 27 \cdot 0.3^{4-1} = 27 \cdot 0.3^3 = 27 \cdot 0.027 = 0.729 \]. So, the fourth term is \(0.729\).

Step 6: Calculate the fifth term

Substitute \(n = 5\) into the formula: \[ a_{5} = 27 \cdot 0.3^{5-1} = 27 \cdot 0.3^4 = 27 \cdot 0.0081 = 0.2187 \]. So, the fifth term is \(0.2187\).

Step 7: Prepare the graph

The first five terms of the sequence are \(27\), \(8.1\), \(2.43\), \(0.729\), and \(0.2187\). Graph these terms against \(n\) values from \(1\) to \(5\) to visually represent the geometric sequence.

Key Concepts

Graphing Sequences

Graphing a geometric sequence helps us visualize how the sequence changes with each term. When graphing the sequence \(a_n = 27 \cdot 0.3^{n-1}\), you plot the sequence terms on the vertical axis (often called the "y-axis") and the term number \(n\) on the horizontal axis (the "x-axis"). For this specific sequence:

• Start by plotting each of the first five terms on the graph. For \(n = 1, 2, 3, 4,\) and \(5\), the terms are \(27, 8.1, 2.43, 0.729,\) and \(0.2187\) respectively.
• The points may appear to decrease rapidly, reflecting the small common ratio of \(0.3\).
• As you graph these points, join them with straight lines to see the trend—a downward, typically curving line confirms the decay of the sequence as \(n\) increases.

Graphing is a great way to quickly understand how each term relates to its position in the sequence and to the common ratio. Protip: You can also use graphing software or an online graphing calculator to plot these points accurately.

Common Ratio

The common ratio in a geometric sequence is a significant factor that determines how each term in the sequence relates to the previous one. In the formula \(a_n = 27 \cdot 0.3^{n-1}\), the common ratio is \(0.3\). Let's understand its importance:

• A common ratio that is less than \(1\) causes the sequence terms to decrease. You notice this in our example where each term is less than the previous term.
• It serves as the multiplier for each subsequent term. For example, the sequence starts at \(27\) and is multiplied by \(0.3\) for each next term, quickly reducing the sequence value.
• If the common ratio was greater than \(1\), each term in the sequence would grow larger than the one before.

Understanding the common ratio helps you predict the behavior of the sequence and anticipate possible future terms. It’s the glue that binds one term to the next in the progression.

Sequence Terms Calculation

Calculating each term in a geometric sequence is made simple through its formula. For the sequence \(a_n = 27 \cdot 0.3^{n-1}\), we'll break down how each first five terms were calculated:

• The term formula, \(a_n = a_1 \cdot r^{n-1}\), uses the first term \(a_1\) and common ratio \(r\).
• **First Term:** For \(n=1\), substitute in the formula: \(27 \cdot 0.3^{0} = 27\). This confirms \(a_1 = 27\).
• **Second Term:** For \(n=2\), calculate as \(27 \cdot 0.3^{1} = 8.1\).
• **Third Term:** For \(n=3\), compute as \(27 \cdot 0.3^{2} = 2.43\).
• **Fourth Term:** For \(n=4\), determine as \(27 \cdot 0.3^{3} = 0.729\).
• **Fifth Term:** Finally, for \(n=5\), find \(27 \cdot 0.3^{4} = 0.2187\).

This systematic process highlights how the formula simplifies finding any term in the sequence swiftly. It's practically a recipe: identify your first ingredient \(a_1\), mix with the common ratio \(r\), and repeat for your desired term number \(n\).