Problem 13 - A Graphical Approach to Precalculus with Limits

Question

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$\frac{4}{5}, 2,5, \frac{25}{2}, \dots$$

Step-by-Step Solution

Answer

\(a_5 = \frac{125}{4}\); \(a_n = \frac{4}{5} \cdot \left(\frac{5}{2}\right)^{n-1}\).

Step 1: Identify the first term

In a geometric sequence, the first term is denoted by \(a_1\). Here, the first term \(a_1\) is \(\frac{4}{5}\).

Step 2: Determine the common ratio

To find the common ratio \(r\) in a geometric sequence, divide the second term by the first term: \(r = \frac{2}{\frac{4}{5}} = \frac{2 \times 5}{4} = \frac{5}{2}\).

Step 3: Use the formula for the nth term

In a geometric sequence, the formula for the nth term \(a_n\) is given by \(a_n = a_1 \cdot r^{n-1}\).

Step 4: Find the 5th term, \(a_5\)

Using the formula: \(a_5 = \frac{4}{5} \cdot \left(\frac{5}{2}\right)^{4}\). Calculate \(\left(\frac{5}{2}\right)^{4} = \frac{625}{16}\). So, \(a_5 = \frac{4}{5} \cdot \frac{625}{16} = \frac{2500}{80} = \frac{125}{4}\).

Step 5: Express \(a_n\)

Using the nth term formula, we have \(a_n = \frac{4}{5} \cdot \left(\frac{5}{2}\right)^{n-1}\). This expression calculates any term in the sequence.

Key Concepts

Common Ratio

In a geometric sequence, understanding the common ratio is crucial for identifying the pattern. The common ratio, often denoted as \( r \), is the factor by which we multiply each term to get the next term in the sequence. For example, using the sequence \( \frac{4}{5}, 2, 5, \frac{25}{2}, \dots \), let's determine the common ratio.
To find this ratio, you simply divide the second term by the first term.

• In our sequence, this calculation is \( r = \frac{2}{\frac{4}{5}} \).
• Performing the division gives us \( r = \frac{5}{2} \).

This common ratio of \( \frac{5}{2} \) tells us that each term is \( \frac{5}{2} \) times the previous term. Identifying the common ratio is your first step in understanding and predicting the terms of any geometric sequence.

nth Term Formula

One of the powerful tools in sequence analysis is the formula for the nth term of a geometric sequence. This formula allows you to find any term in the sequence without starting from the first term and multiplying repeatedly. The nth term formula is given by:

• \( a_n = a_1 \cdot r^{n-1} \)
• Here, \( a_1 \) is the first term and \( r \) is the common ratio.

For our specific sequence, \( a_1 = \frac{4}{5} \) and \( r = \frac{5}{2} \). Plugging these values into the formula,

• We get \( a_n = \frac{4}{5} \cdot \left( \frac{5}{2} \right)^{n-1} \).

This formula is especially helpful for calculating the fifth term \( a_5 \) without manually going through each term.
Using the nth term formula can save time and reduce errors when dealing with large sequences or large values of \( n \).

Sequence Analysis

Analyzing a sequence involves more than just finding terms; it includes understanding the behavior and characteristics of the sequence. Let's use our sequence example: \( \frac{4}{5}, 2, 5, \frac{25}{2}, \dots \).

• The first step in sequence analysis is to check if the sequence is geometric, which occurs if there is a consistent common ratio across terms.
• Another aspect to consider is the growth or decay of the sequence. A common ratio greater than one, as in \( \frac{5}{2} \), indicates exponential growth.

By analyzing the sequence, we can predict future terms and understand patterns in the data. It’s crucial when sequences model real-world scenarios, such as population growth, financial investments, or physics problems.
This methodical approach helps in building a deep comprehension of sequences and applies mathematical reasoning to practical contexts.