Problem 25 - Precalculus: Mathematical for Calculus

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 nth term is \(a_n = 0.3 \times (-0.3)^{n-1}\).

Step 1: Identify the Common Ratio

To find the common ratio of a geometric sequence, divide the second term by the first term: \(-0.09 \div 0.3 = -0.3\). The common ratio \(r = -0.3\). This means each term is \(-0.3\) times the previous term.

Step 2: Calculate the Fifth Term

Using the formula for the geometric sequence, \(a_n = a_1 r^{n-1}\), where \(a_1 = 0.3\), \(r = -0.3\), and \(n = 5\), calculate the fifth term: \(a_5 = 0.3 \times (-0.3)^{4} = 0.3 \times 0.0081 = 0.00243\).

Step 3: Find the General Formula for the nth Term

Using the geometric sequence formula \(a_n = a_1 r^{n-1}\), substitute the known values \(a_1 = 0.3\) and \(r = -0.3\) to write the formula: \(a_n = 0.3 \times (-0.3)^{n-1}\). This formula can be used to find any term in the sequence.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a critical component. It determines how each term relates to the one before it. The common ratio is the factor by which you multiply one term to get to the next. To find the common ratio, you simply divide the second term of the sequence by the first term. For example, in the sequence 0.3, -0.09, 0.027, -0.0081, ... the common ratio would be calculated as: \[ \text{Common ratio} = \frac{-0.09}{0.3} = -0.3 \] This tells us that each number in the sequence is multiplied by -0.3 to get the next term.

• If the common ratio is positive, all terms will have the same sign.
• If the common ratio is negative, the terms will alternate in sign.

Recognizing the common ratio can help you predict future terms and understand the behavior of the sequence.

Utilizing the nth Term Formula

The nth term formula is a powerful tool in geometry sequences. It provides a way to find any term in the sequence without listing all the preceding ones. The formula is expressed as: \[ a_n = a_1 \cdot r^{n-1} \] Here's how you can use it: - **\(a_1\)**: This is the first term of the sequence.- **\(r\)**: Represents the common ratio, which you multiply with successive terms.- **\(n\)**: Indicates the position of the term in the sequence. So, given the first term as 0.3 and the common ratio as -0.3, you can find any term by substituting into the nth term formula. For example, the 5th term can be calculated as:\[ a_5 = 0.3 \times (-0.3)^{4} = 0.00243 \] This method is straightforward and efficient, especially for large sequences.

Calculating Sequence Terms

Calculating specific terms in a geometric sequence is made easy using the common ratio and the nth term formula. When asked to find a certain term, like the fifth term, follow these steps:1. **Identify your starting point**: Know your first term, which is often given. Here, it's 0.3.2. **Use the common ratio**: Multiply the first term by the common ratio enough times to reach your desired term.3. **Apply the nth term formula**:\[ a_n = 0.3 \times (-0.3)^{n-1} \] replacing \(n\) with the term position you are looking for. For the fifth term calculation, this was:\[ a_5 = 0.3 \times (-0.3)^{4} = 0.00243 \]Each calculation involves basic arithmetic and exponentiation. The order of operations is crucial for accuracy:

• First, calculate the power of the common ratio.
• Second, multiply it by the initial term.

Understanding this process can make solving these problems straightforward and less daunting for students.