Problem 11 - Precalculus: Mathematics 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: Identify the formula

To find the nth term of a geometric sequence, we use the formula for the nth term: \( a_n = a \times r^{n-1} \). Here, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 2: Substitute given values

We are given \( a = \frac{5}{2} \) and \( r = -\frac{1}{2} \). We need to find the fourth term, so \( n = 4 \). Substitute these values into the formula: \( a_4 = \frac{5}{2} \times \left(-\frac{1}{2}\right)^{4-1} \).

Step 3: Calculate the power of the ratio

Calculate \( \left(-\frac{1}{2}\right)^{3} \). Since the exponent is odd, the result will be negative. Calculate \( \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \), which equals \( -\frac{1}{8} \).

Step 4: Multiply to find the fourth term

Now, take the result from Step 3 and multiply it with the first term: \( a_4 = \frac{5}{2} \times -\frac{1}{8} \). Simplify the multiplication: \( \frac{5}{2} \times -\frac{1}{8} = -\frac{5}{16} \).

Key Concepts

nth term

The nth term in a geometric sequence is a specific term that we want to identify or calculate within the sequence. To find it, we use a simple formula:

• \( a_n = a \times r^{n-1} \)

Here, \( a \) is the first term; \( r \) is the common ratio, which we'll explain in detail later, and \( n \) is the term number that you are trying to find.

For example, if you are asked to find the 4th term and you know the first term \( a \) and the common ratio \( r \), you plug these values into the formula to find \( a_4 \). For the problem given, we know \( a = \frac{5}{2} \), \( r = -\frac{1}{2} \), and you want the 4th term, so \( n = 4 \). You would calculate \( a_4 = \frac{5}{2} \times \left(-\frac{1}{2}\right)^{4-1} \).

This formula is helpful because it allows you to determine any term in the sequence without listing all previous terms.

common ratio

The common ratio is a crucial part of understanding geometric sequences. It is a constant used to multiply each term to get to the next term in the sequence. In mathematical terms, it's represented by the symbol \( r \).

• If the common ratio is greater than 1, the sequence will grow rapidly.
• If it's between 0 and 1, each term will get progressively smaller.
• If \( r \) is negative, the terms will alternate in sign.

In our example, the common ratio \( r = -\frac{1}{2} \) means each term is half of the previous term but with the opposite sign.

To see this in action, consider the sequence \( \frac{5}{2}, -\frac{5}{4}, \frac{5}{8}, ... \). You get each term by multiplying the preceding term by \(-\frac{1}{2}\). This simple concept helps predict the behavior of the sequence as it progresses.

geometric series formula

Besides finding specific terms, sometimes you need the sum of multiple terms in a geometric sequence, like in a geometric series. To calculate the sum of the first \( n \) terms in a geometric sequence, we use this formula:

• \( S_n = a \frac{1-r^n}{1-r} \)

This formula is applicable when \( |r| < 1 \) and \( r eq 1 \). See how it uses the first term \( a \), the common ratio \( r \), and the number of terms \( n \).

For example, if you wanted to find the sum of the first four terms of the sequence provided in the problem where \( a = \frac{5}{2} \) and \( r = -\frac{1}{2} \), you would plug those values into the formula as follows:

\[ S_4 = \frac{5}{2} \frac{1-(-\frac{1}{2})^4}{1-(-\frac{1}{2})} \]
This formula is very helpful in solving problems that require adding up elements in a geometric sequence.