Problem 62 - Precalculus with Limits

Question

In Exercises 61 - 66, use a graphing utility to graph the first 10 terms of the sequence. \( a_n 10\left(1.5\right)^{n - 1} \)

Step-by-Step Solution

Answer

The graphed sequence will show an exponential increase due to the nature of the formula \( a_n = 10(1.5)^{n - 1} \). The points on the graph representing the first ten terms of the sequence will illustrate this.

Step 1: Compute the first 10 terms

This step involves utilizing the given formula \( a_n = 10(1.5)^{n - 1} \) to find the first ten terms in the sequence. Using this formula, we will substitute the values of n from 1 to 10, and calculate each corresponding \( a_n \).

Step 2: Graph the sequence

After obtaining the first 10 terms, the next step is to graph the sequence. On the x-axis, use values from 1 to 10 representing the position of the term in the sequence. For the y-axis, use the values of \( a_n \) obtained from step 1. Each point on the graph should represent a term in the sequence, with the x-coordinate corresponding to the term position and the y-coordinate corresponding to the term's value.

Step 3: Interpreting the graph

After graphing the sequence, the trend of the sequence can be analyzed. For this specific sequence, an exponential growth can be expected, given the exponential nature of the formula \( a_n = 10(1.5)^{n - 1} \).

Key Concepts

Exponential Growth

Exponential growth describes a situation where values increase rapidly over time, following a consistent rate. In the formula given, \( a_n = 10(1.5)^{n - 1} \), the term \( 1.5 \) functions as the growth factor. Whenever this factor is greater than 1, it signals exponential growth. This means that each subsequent term in the sequence is 1.5 times the previous one.

In exponential growth:

• The initial value is crucial. Here, it's 10, which determines where the growth starts.
• The position \( n \) controls how many times the growth factor is applied.
• As \( n \) increases, the terms grow increasingly larger at an accelerating rate.

If you graph such a sequence, you'll notice a curve that starts slowly and then sharply increases. This shape shows the power of exponential growth—small changes at first, but massive results over time.

Graphing Sequences

Graphing sequences is an essential way to visualize and understand how terms change over time. For our sequence \( a_n = 10(1.5)^{n - 1} \), plotting each term on a graph can help us see its exponential growth.

When graphing:

• The x-axis represents the term position \( n \). This axis simply goes from 1 to 10 for our case.
• The y-axis represents the value of each term \( a_n \). These are the results from our earlier calculations.
• Mark each point on the graph where the x-coordinate is \( n \) and the y-coordinate is the value of \( a_n \).

After plotting all 10 terms, you'll have a series of points rising steeply, reflecting the exponential growth. These visuals not only confirm calculated values but also assist in predicting future terms and seeing how rapid growth evolves.

Term Calculation

Calculating terms in a sequence involves substituting values into a given formula to find each specific term. Here, with the formula \( a_n = 10(1.5)^{n - 1} \), we're determining the first ten terms of the sequence.

Steps for term calculation:

• Substitute \( n = 1 \) into the formula. This gives us \( 10(1.5)^0 = 10 \). Hence, the first term is 10.
• Continue this substitution process for \( n = 2, 3, \ldots, 10 \), calculating each term.
• For instance, when \( n = 2 \), the term is \( 10(1.5)^1 = 15 \).
• Each subsequent term involves multiplying the previous term by 1.5.

The consistent approach ensures accuracy and reinforces how exponential growth functions. The method of calculation becomes faster and more intuitive as familiarity with the sequence structure grows.