Problem 1 - A Graphical Approach to Precalculus with Limits

Question

CHECKING ANALYTIC SKILLS Write the terms of the geometric sequence that satisfies the given conditions. Do not use a calculator. $$a_{1}=\frac{5}{3}, r=3, n=4$$

Step-by-Step Solution

Answer

The terms of the sequence are: \( \frac{5}{3}, 5, 15, 45 \).

Step 1: Understand the Geometric Sequence Formula

A geometric sequence can be defined using the formula for the nth term: \( a_n = a_1 \cdot r^{n-1} \), 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 is given as \( a_1 = \frac{5}{3} \). So, the first term of the sequence is \( \frac{5}{3} \).

Step 3: Calculate the Second Term

Use the formula with \( n=2 \):\[ a_2 = \frac{5}{3} \cdot 3^{1} = \frac{5}{3} \cdot 3 = 5 \].Thus, the second term is 5.

Step 4: Calculate the Third Term

Use the formula for \( n=3 \):\[ a_3 = \frac{5}{3} \cdot 3^{2} = \frac{5}{3} \cdot 9 = 15 \].The third term is 15.

Step 5: Calculate the Fourth Term

Use the formula for \( n=4 \):\[ a_4 = \frac{5}{3} \cdot 3^{3} = \frac{5}{3} \cdot 27 = 45 \].The fourth term is 45.

Key Concepts

Sequence Formula

In mathematics, a sequence is an ordered list of numbers. A geometric sequence is a special type of sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula to find any term in a geometric sequence is essential. This equation, known as the sequence formula, is expressed as:\[ a_n = a_1 \cdot r^{n-1} \]Where:

• \(a_n\) is the nth term you want to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the position of the term in the sequence (like 1st, 2nd, 3rd, etc.).

Understanding this formula allows you to identify any term in the sequence if you know the first term and the common ratio. It opens the doorway to analyzing patterns and predicting further numbers in the series, which is particularly useful in various scientific and economic applications.

Common Ratio

The common ratio \(r\) is a pivotal element in geometric sequences. It is the factor by which you multiply one term to get to the next. Knowing the common ratio helps you understand how quickly the sequence grows or diminishes.To determine the common ratio, divide any term in the sequence by the preceding term. For instance, in a sequence starting with \(a_1 = \frac{5}{3}\) and a common ratio of \(3\), we can calculate subsequent terms by multiplying each by 3:- For the first few terms: - The first term is \(\frac{5}{3}\) - The second term is \(\frac{5}{3} \cdot 3 = 5\) - The third term is \(5 \cdot 3 = 15\) - The fourth term is \(15 \cdot 3 = 45\)Understanding the role of the common ratio helps in predicting the behavior of the sequence, whether it will increase to large numbers, decrease rapidly, or remain relatively stable.

Nth Term Calculation

The ability to calculate the nth term in a geometric sequence is necessary for extending the sequence beyond the initial terms. This involves using the sequence formula. Let's break it down with a practical example.Given:\(a_1 = \frac{5}{3}\), \(r = 3\), and you want the 4th term.Using the formula \( a_n = a_1 \cdot r^{n-1} \):For the 4th term:\[ a_4 = \frac{5}{3} \cdot 3^{4-1} = \frac{5}{3} \cdot 3^3 = \frac{5}{3} \cdot 27 = 45\]Breaking it down into simpler steps:

• First, calculate the exponent: \(3^3 = 27\)
• Then, multiply the result by the first term: \(\frac{5}{3} \cdot 27\)
• Simplify to get the term value \(45\)

By understanding how to find the nth term, you can extend any geometric sequence to the desired level, offering insights, patterns, and predictions for practical use.