Problem 66 - Precalculus with Limits

Question

In Exercises 61 - 66, use a graphing utility to graph the first 10 terms of the sequence. \( a_n 10\left(1.2\right)^{n - 1} \)

Step-by-Step Solution

Answer

The first 10 terms of the sequence can be calculated using the formula \( a_n = ar^{n - 1} \). The values obtained can then be graphed using any standard graphing utility. The resulting graph should illustrate a geometric sequence, where each consequent term is 1.2 times larger than the previous term.

Step 1: Identifying the Type of Sequence

The provided sequence, \( a_n = 10\left(1.2\right)^{n - 1} \) is a geometric sequence, where 10 is the first term (a), 1.2 is the common ratio (r), and n is the term number.

Step 2: Calculation of First 10 Terms

Use the formula \( a_n = ar^{n - 1} \) to calculate each of the first ten terms. For the first term (n=1), the calculation will be \( 10(1.2)^{1 - 1} = 10 \). For the second term (n=2), the calculation will be \( 10(1.2)^{2 - 1} = 12 \), and so on until the tenth term.

Step 3: Graphing the Terms

Use a graphing utility to plot the computed values on a graph. The term number (n) will be represented along the x-axis, and the corresponding term value (computed in step 2) will be represented along the y-axis.

Key Concepts

Understanding the Common Ratio in a Geometric Sequence

In the geometric sequence given by the formula \( a_n = 10(1.2)^{n-1} \), the common ratio \( r \) plays a crucial role. The common ratio is the factor by which we multiply each term to get the next one in the sequence.
This particular sequence has a common ratio of 1.2. This means that every term is 1.2 times larger than the previous term. For example, from the first term (10) to the second term (12), we can see this multiplication in action:

• First term: \( a_1 = 10 \)
• Second term: \( a_2 = 10 \times 1.2 = 12 \)

This pattern continues as we move through the sequence, illustrating the repetitive multiplying process that defines geometric sequences. Understanding the common ratio is key to unlocking the sequence's behavior.

What is the Term Number?

In a geometric sequence like \( a_n = 10(1.2)^{n-1} \), the term number \( n \) is an essential component. It represents the position of a term within the sequence. It's common to start with \( n = 1 \), which gives us the first term.
Knowing the term number helps you identify which term you're working with or need to calculate. For instance, to find the third term, you set \( n = 3 \). Each increase in the term number typically reflects moving from one term to the next:

• First term: \( n = 1 \)
• Second term: \( n = 2 \)
• Third term: \( n = 3 \)

In our exercise, the term number also plays a critical role in graphing, as we'll plot term numbers along the x-axis, helping visualize the sequence's growth.

Utilizing a Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical sequences and data, making it easier to understand complex relationships. For the sequence \( a_n = 10(1.2)^{n-1} \), a graphing utility can help plot each term value against its term number.
To graph the first ten terms of this sequence, you'll:

• Start by calculating each term using the formula \( a_n = 10(1.2)^{n-1} \), from \( n=1 \) to \( n=10 \).
• Next, plot these values with term number \( n \) on the x-axis and the calculated term value on the y-axis.
• The result is a visual graph showing how quickly the sequence grows, thanks to the common ratio of 1.2.

The graph provides an immediate visual cue, clarifying how each term is progressively larger, highlighting the exponential nature of the geometric growth.

The Significance of the First Term

In any geometric sequence, the first term \( a \) is a foundational component. For the given sequence \( a_n = 10(1.2)^{n-1} \), this first term is \( a = 10 \).
The first term sets the starting point from which the sequence begins its progression. It's the baseline that gets multiplied by the common ratio to find subsequent terms. For example, from this initial term, every other term is calculated using the common ratio of 1.2, showing the growth path of the sequence.
The importance of the first term also comes into play when graphing. On the graph, the first term determines the starting point of the curve or plot on the y-axis at \( n=1 \). It gives a clear representation of where the sequence starts before rapidly increasing due to the common ratio.