Problem 25 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 0.3,-0.09,0.027,-0.0081, \ldots $$

Step-by-Step Solution

Answer

The common ratio is \(-0.3\), the fifth term is 0.00243, and the \(n\)th term is \( a_n = 0.3 \cdot (-0.3)^{n-1} \).

Step 1: Identify the Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a constant known as the common ratio. To find this ratio, divide the second term by the first term: \( \frac{-0.09}{0.3} = -0.3 \). Therefore, the common ratio \( r \) is \(-0.3\).

Step 2: Calculate the Fifth Term

The fifth term can be determined using the formula for the \(n\)th term of a geometric sequence: \( a_n = a_1 \cdot r^{n-1} \). Here, \( a_1 = 0.3 \), \( r = -0.3 \), and \( n = 5 \). Substitute these values to find the fifth term: \( a_5 = 0.3 \cdot (-0.3)^{4} = 0.3 \cdot 0.0081 = 0.00243 \). Hence, the fifth term is 0.00243.

Step 3: Find the Formula for the \(n\)th Term

Using the formula for the \(n\)th term of a geometric sequence, \( a_n = a_1 \cdot r^{n-1} \), and knowing that \( a_1 = 0.3 \) and \( r = -0.3 \), we can express the \(n\)th term as \( a_n = 0.3 \cdot (-0.3)^{n-1} \). This formula allows us to find any term in the sequence.

Key Concepts

Common Ratio

In any geometric sequence, the concept of the **common ratio** is fundamental. This ratio represents the factor by which we multiply each term to obtain the next one in the sequence. To find it, you take a term and divide it by the previous term.
For example, in the sequence 0.3, -0.09, 0.027, -0.0081, each term is obtained by multiplying the previous term by the same number. Here, the common ratio is calculated by taking the second term and dividing it by the first term, which is \( \frac{-0.09}{0.3} = -0.3 \).
This means that every term is -0.3 times the one before it. Understanding the common ratio helps us to easily predict future terms in the sequence, making it a powerful tool in sequence calculations.

Fifth Term

The **fifth term** of a geometric sequence can be calculated using its starting term and the common ratio. We use the **nth term formula** specific to geometric sequences: \( a_n = a_1 \cdot r^{n-1} \).
Here, the first term \( a_1 \) is 0.3, the common ratio \( r \) is -0.3, and we are interested in the fifth term, so \( n = 5 \). Substituting in these values, the formula becomes:

• \( a_5 = 0.3 \cdot (-0.3)^{4} \)
• \( a_5 = 0.3 \cdot 0.0081 \)
• \( a_5 = 0.00243 \)

Hence, the fifth term is 0.00243.
This formula efficiently guides us to find any term, not just the fifth, provided the first term and the common ratio are known.

N-th Term Formula

The **n-th term formula** is a key concept in understanding geometric sequences. It provides a way to find any term in the sequence without listing all previous terms. The general formula for the nth term of a geometric sequence is given by:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Where:

• \( a_n \) is the n-th term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number.

In our example, \( a_1 \) is 0.3 and \( r \) is -0.3. Therefore, for any \( n \), the term can be calculated using:
\( a_n = 0.3 \cdot (-0.3)^{n-1} \).
This expression allows you to plug in any value for \( n \) to find the corresponding term without going through all the previous ones. This is incredibly useful, especially when dealing with large sequences.