Problem 10 - Algebra and Trigonometry

Question

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left.s_{n}\right\\}=\left\\{(-5)^{n}\right\\} $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of \ -5 \ and the first four terms are \ -5, 25, -125, 625.

Step 1: Identify the general form of the sequence

The given sequence is \( s_{n} = \{(-5)^n\} \). This means the nth term of the sequence is \( (-5)^n \).

Step 2: Determine the common ratio

To determine if the sequence is geometric, calculate the ratio between consecutive terms. The ratio can be found by dividing each term by its preceding term. For example, the ratio between the second term \( s_2 = (-5)^2 \) and the first term \( s_1 = (-5)^1 \) is \[ \frac{s_2}{s_1} = \frac{(-5)^2}{(-5)^1} = \frac{25}{-5} = -5 \].

Step 3: Verify the common ratio for other terms

Check the common ratio for the next terms. \( s_3 = (-5)^3 \) and \( s_2 = (-5)^2 \), \[ \frac{s_3}{s_2} = \frac{(-5)^3}{(-5)^2} = \frac{-125}{25} = -5 \]. It confirms the ratio is constant (\(-5\)).

Step 4: List the first four terms of the sequence

Calculate the first four terms using the general form \( (-5)^n \): \ s_1 = (-5)^1 = -5 \ \ s_2 = (-5)^2 = 25 \ \ s_3 = (-5)^3 = -125 \ \ s_4 = (-5)^4 = 625 \.

Key Concepts

Common Ratio

In a geometric sequence, the **common ratio** is the factor that each term in the sequence is multiplied by to get the next term. The common ratio is vital because it defines how the sequence progresses. To determine the common ratio, divide any term in the sequence by the term that comes before it. For instance, consider the sequence given by \{(-5)^n\}. To find the common ratio, divide the second term \( s_2 \) by the first term \( s_1 \): \(\frac{s_2}{s_1} = \frac{25}{-5} = -5 \). Repeating this for later terms confirms the ratio remains constant: \(\frac{s_3}{s_2} = \frac{-125}{25} = -5 \). Thus, the common ratio for this sequence is \( r = -5 \). Understanding the common ratio helps in predicting and calculating the subsequent terms in any geometric sequence.

Sequence Terms

The **terms** of a geometric sequence follow a specific pattern based on the common ratio. For a sequence defined by \( (-5)^n \), the terms can be individually computed using the general form. The first four terms are calculated as follows:

• First term (\(s_1\)): \( s_1 = (-5)^1 = -5 \)
• Second term (\(s_2\)): \( s_2 = (-5)^2 = 25 \)
• Third term (\(s_3\)): \( s_3 = (-5)^3 = -125 \)
• Fourth term (\(s_4\)): \( s_4 = (-5)^4 = 625 \)

As seen, each term is calculated by raising \( -5 \) to the power of its position number. Recognizing how each term derives from the previous ones through the common ratio \( r = -5 \) is key to understanding and analysing the pattern of any geometric sequence.

General Form

The **general form** of a geometric sequence conveys a formula to find any term in the sequence. For the given sequence \[\left.s_{n}=(-5)^n\right\], the general form is \( s_n = (-5)^n \). Here, \( n \) represents the position of the term in the sequence, and \( (-5) \) is the base being raised to the power \( n \). This general form is fundamental because:

• It provides a direct way to find any term without listing all previous terms.
• Using the general form formula, one can quickly determine any term like \( s_1 \) to \( s_n \).
• It helps in identifying the properties and behavior of the sequence, such as exponential growth or decay depending on the base and the position \( n \).

Understanding the general form and how it translates into the calculation of each term is essential for mastering geometric sequences.