Problem 7 - Precalculus: Mathematical for Calculus

Question

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=\frac{5}{2}, \quad r=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The fourth term is \(-\frac{5}{16}\).

Step 1: Understand the formula for the nth term

The nth term of a geometric sequence can be calculated using the formula: \( a_n = a imes r^{(n-1)} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number to be calculated.

Step 2: Substitute values for the fourth term

In this case, to find the fourth term, we will substitute \( a = \frac{5}{2} \), \( r = -\frac{1}{2} \), and \( n = 4 \) into the formula: \( a_4 = \frac{5}{2} \times \left( -\frac{1}{2} \right)^{3} \).

Step 3: Calculate power of the common ratio

Calculate \( \left( -\frac{1}{2} \right)^{3} \). Since the power is odd, the result is negative: \( \left( -\frac{1}{2} \right)^3 = -\frac{1}{8} \).

Step 4: Multiply with the first term

Now, multiply \( \frac{5}{2} \) by \( -\frac{1}{8} \): \( a_4 = \frac{5}{2} \times -\frac{1}{8} = -\frac{5}{16} \).

Key Concepts

Understanding the nth Term Formula

In a geometric sequence, the nth term formula is essential. It helps us find any term in the sequence quickly without listing all the terms. The formula is given by: \[ a_n = a \times r^{(n-1)} \]Here,

• \(a_n\) is the nth term we want to find.
• \(a\) is the first term in the sequence.
• \(r\) represents the common ratio.
• \(n\) is the position of the term we are looking for.

For instance, if we want the fourth term and we know the first term and the common ratio, we simply substitute these values along with \(n = 4\) into the formula. This approach simplifies the process and saves time, especially with larger sequences.

Identifying the First Term

The first term of a geometric sequence is indicated by \(a\). It's crucial because it sets the starting point for the sequence. Here, the first term is part of what makes each term unique based on its position. For example, in the sequence from our problem, the first term \(a\) is \(\frac{5}{2}\). This first term influences the whole sequence since each subsequent term is obtained by multiplying the previous term by the common ratio. In fact, without the first term, we couldn't even begin constructing our sequence. Therefore, knowing this initial value is key to finding any term in the sequence.

Understanding the Common Ratio

The common ratio, denoted as \(r\), is the constant factor between consecutive terms of a geometric sequence. It tells us how the sequence grows or shrinks. In our example, the common ratio is \(-\frac{1}{2}\).

• A positive common ratio results in terms with the same sign as the first term.
• A negative common ratio results in alternating signs for successive terms.

This consistent multiplication by \(r\) defines the entire series. It's why geometric sequences can vary widely, from rapidly escalating sequences to those that shrink to zero. Understanding this concept is crucial, as it not only aids computation but also gives insight into the sequence's behavior as it progresses.