Problem 1 - A Graphical Approach to College Algebra

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 sequence is \(\frac{5}{3}, 5, 15, 45\).

Step 1: Understand the Problem

We need to find the first four terms of a geometric sequence given the first term \(a_1 = \frac{5}{3}\) and the common ratio \(r = 3\). We are asked to find the sequence up to \(n = 4\).

Step 2: Apply the Formula for the nth Term

In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio. The nth term of a geometric sequence is given by the formula \(a_n = a_1 \times r^{(n-1)}\).

Step 3: Calculate the First Term

The first term is already provided as \(a_1 = \frac{5}{3}\).

Step 4: Calculate the Second Term

Using the formula, the second term \(a_2 = a_1 \times r^{(2-1)} = \frac{5}{3} \times 3 = 5\).

Step 5: Calculate the Third Term

The third term is calculated as \(a_3 = a_1 \times r^{(3-1)} = \frac{5}{3} \times 3^2 = \frac{5}{3} \times 9 = 15\).

Step 6: Calculate the Fourth Term

The fourth term is \(a_4 = a_1 \times r^{(4-1)} = \frac{5}{3} \times 3^3 = \frac{5}{3} \times 27 = 45\).

Key Concepts

Common Ratio

The common ratio is an essential element of any geometric sequence. It is the factor by which we multiply each term to get to the next one in the sequence. In other words, it dictates the growth pattern of the sequence. For the given sequence, the common ratio is 3. This means each term is three times larger than the one before it.
To find this ratio, if not provided, you can divide any term by the preceding term:

• Formula: \( r = \frac{a_{n}}{a_{n-1}} \)

This calculation will consistently yield the common ratio in a geometric progression. Understanding the common ratio helps in not only constructing the sequence but also predicting future terms.

Geometric Progression

Geometric progression, often termed as a geometric sequence, is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This progression is characterized by the constant multiplication factor, which significantly differs from arithmetic sequences, where a constant addition forms the progression.
For example, in our earlier sequence

• First term: \(a_1 = \frac{5}{3}\)
• Common ratio: \(r = 3\)
• Second term: \(a_2 = 5\)

Each of these terms is part of the geometric progression we have identified. Understanding this structure helps in predicting subsequent terms or calculating specific terms using the nth term formula without generating all preceding terms.

nth Term Formula

The nth term formula in a geometric sequence allows us to find any term within the sequence without listing all previous terms. This formula is a mathematical shortcut to compute desired terms efficiently. The formula for the nth term of a geometric sequence is:

• \(a_n = a_1 \times r^{(n-1)}\)

Where:

• \(a_n\) is the term you want to find,
• \(a_1\) is the first term, and
• \(r\) is the common ratio.

For instance, to find the fourth term in our sequence, substitute the values into the formula: \(a_4 = \frac{5}{3} \times 3^{3} = 45\). This method is concise and avoids unnecessary calculations, paving the way for understanding large sequences effortlessly.