Problem 26 - Precalculus with Limits

Question

In Exercises 17 - 28, write the first five terms of the geometric sequence \( a_1 = 4, r = -\dfrac{1}{\sqrt{2}} \)

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are 4, \(-2\sqrt{2}\), -2, \(\sqrt{2}\), 1

Step 1: Identify and setup

Identify the first term and the common ratio from the given problem. The first term \(a_1\) is 4 and the ratio \(r\) is \(-\dfrac{1}{\sqrt{2}}\). The term \(a_n\) in a geometric sequence can be found using the formula \(a_n = a_1 \cdot r^{(n-1)}\).

Step 2: First term (n=1)

First, leave \(n\) as 1 to find the first term. Substituting into the formula gives us: \(a_1 = 4 \cdot (- \frac{1}{\sqrt{2}} )^{(1-1)} = 4\)

Step 3: Second term (n=2)

Next, substitute \(n\) by 2 to find the second term. Substituting into the formula gives us: \(a_2 = 4 \cdot (- \frac{1}{\sqrt{2}} )^{(2-1)} = -\dfrac{4}{\sqrt{2}} = -2\sqrt{2}\)

Step 4: Third term (n=3)

Next, substitute \(n\) by 3 to find the third term. Substituting into the formula gives us: \(a_3 = 4 \cdot (- \frac{1}{\sqrt{2}} )^{(3-1)} = -2\)

Step 5: Fourth term (n=4)

Next, substitute \(n\) by 4 to find the fourth term. Substituting into the formula gives us: \(a_4 = 4 \cdot (- \frac{1}{\sqrt{2}} )^{(4-1)} = \sqrt{2}\)

Step 6: Fifth term (n=5)

Finally, substitute \(n\) by 5 to find the fifth term. Substituting into the formula gives us: \(a_5 = 4 \cdot (- \frac{1}{\sqrt{2}} )^{(5-1)} = 1\)

Key Concepts

Common Ratio

In a geometric sequence, every term is a product of the previous term multiplied by a fixed number known as the **common ratio**. This ratio remains consistent throughout the entire sequence.
For instance, in the given problem, the common ratio \(r\) is \(-\frac{1}{\sqrt{2}}\). This means each term is derived by multiplying the previous term by this ratio. Therefore, the series will flip signs in alternating terms because the common ratio is negative.

• First, determine the common ratio from the sequence definition.
• Understand that a negative ratio indicates a flip in sign across terms.

Knowing the common ratio allows us to predict the behavior and direction of the sequence, which in this case, will alternate between positive and negative terms.

First Term

The **first term** in a geometric sequence is often denoted by \(a_1\) and serves as the starting point for generating the subsequent terms.
In the problem at hand, the first term is provided as **4**.
Let's see why the first term is crucial:

• The first term provides the initial value, which every other term is derived from.
• It is directly substituted into the sequence formula for calculations.

Thus, understanding the first term is fundamental in laying the groundwork for building the entire sequence and helps anchor the sequence's values.

Sequence Formula

In mathematics, the formula for a **geometric sequence** is crucial for determining any term within the sequence. The formula is given by \[a_n = a_1 \cdot r^{(n-1)}\]where:

• \(a_n\) is the nth term of the sequence.
• \(a_1\) is the first term.
• \(r\) is the common ratio.
• \(n\) represents the term's position in the sequence.

By plugging the values of the first term and common ratio, along with the desired term's position \(n\), the nth term can be easily calculated.
This formula highlights the exponential growth or decay observed in geometric sequences, which differs them from arithmetic sequences. It is an efficient tool to predict any term without listing all preceding terms.

Mathematics

**Mathematics** lays the foundation for understanding structures and patterns, with geometric sequences being a vivid illustration. They showcase how exponential relationships function in formulas and real-world phenomena.
Engaging with sequences such as geometric ones empower students to:

• Recognize patterns and derive formulas from given data.
• Use algebra to manipulate and solve complex problems.
• Understand how ratios and exponential changes relate to sequences.

Mathematical concepts, including those seen in geometric sequences, enhance our comprehension of rational progression and aid in developing critical problem-solving skills. These skills are valuable beyond the classroom in fields like finance, science, and engineering.