Problem 11 - Algebra and Trigonometry

Question

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{a_{n}\right\\}=\left\\{-3\left(\frac{1}{2}\right)^{n}\right\\} $$

Step-by-Step Solution

Answer

Common ratio: \(\frac{1}{2}\); First four terms: -3, \(-\frac{3}{2}\), \(-\frac{3}{4}\), \(-\frac{3}{8}\).

Step 1: Identify the general term

Given the sequence \(\{a_{n}\} = \{-3\left(\frac{1}{2}\right)^{n}\}\), the general term is \(a_{n} = -3\left(\frac{1}{2}\right)^{n}\).

Step 2: Generate the first four terms

Substitute \(n = 0, 1, 2, 3\) into the general term to find the first four terms.\left.\begin{array}{l}a_{0} = -3\left(\frac{1}{2}\right)^{0} = -3\left(1\right) = -3 \a_{1} = -3\left(\frac{1}{2}\right)^{1} = -3\left(\frac{1}{2}\right) = -\frac{3}{2} \a_{2} = -3\left(\frac{1}{2}\right)^{2} = -3\left(\frac{1}{4}\right) = -\frac{3}{4} \a_{3} = -3\left(\frac{1}{2}\right)^{3} = -3\left(\frac{1}{8}\right) = -\frac{3}{8} \end{array}\right.

Step 3: Determine the common ratio

The common ratio \(r\) is calculated as the ratio of successive terms.\left.\begin{array}{l}r = \frac{a_{1}}{a_{0}} = \frac{-\frac{3}{2}}{-3} = \frac{1}{2} \r = \frac{a_{2}}{a_{1}} = \frac{-\frac{3}{4}}{-\frac{3}{2}} = \frac{1}{2} \r = \frac{a_{3}}{a_{2}} = \frac{-\frac{3}{8}}{-\frac{3}{4}} = \frac{1}{2} \end{array}\right.The common ratio is consistently \(\frac{1}{2}\).

Step 4: Confirm the sequence is geometric

Since the ratio between all successive terms is the same and equal to \(\frac{1}{2}\), the sequence is geometric.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the factor by which you multiply each term to get the next term. For the given sequence \{-3\left(\frac{1}{2}\right)^{n}\}, the common ratio \(r\) can be found by dividing any term by its preceding term. Let's look at the terms: when dividing \(a_1\) by \(a_0\), \(r = \frac{-\frac{3}{2}}{-3} = \frac{1}{2}\). You can see that this process gives the same result consistently. Thus, the common ratio for this sequence is \(\frac{1}{2}\). Understanding the common ratio is crucial for analyzing geometric sequences.

Sequence Terms

To understand geometric sequences better, let's generate the first few terms. For the sequence \(\{a_{n}\} = \{-3\left(\frac{1}{2}\right)^{n}\}\), we need to substitute \(n\) with 0, 1, 2, and 3.
When \(n = 0\), \(a_0 = -3\times1 = -3\). When \(n = 1\), \(a_1 = -3\left(\frac{1}{2}\right) = -\frac{3}{2}\). When \(n = 2\), \(a_2 = -3\left(\frac{1}{4}\right) = -\frac{3}{4}\). When \(n = 3\), \(a_3 = -3\left(\frac{1}{8}\right) = -\frac{3}{8}\). Analyzing these terms gives a better understanding of how the sequence behaves. Each term is just the previous term multiplied by the common ratio \(\frac{1}{2}\).

Geometric Progression

A geometric progression is a sequence where each term is obtained by multiplying the previous term by a fixed non-zero number called the common ratio. To confirm if a given sequence is geometric, you need to check: Does the ratio \(\frac{a_{n+1}}{a_n}\) remain constant?
For our sequence \(\{a_{n}\} = \{-3\left(\frac{1}{2}\right)^{n}\}\), the ratio is consistently \(\frac{1}{2}\). By substituting different values of \(n\), we verified that the ratio between successive terms is always the same. Hence, this sequence is geometric. Understanding this helps in recognizing patterns and solving problems related to geometric sequences.