Problem 17 - Contemporary Precalculus

Question

In Exercises \(13-22,\) one term and the common ratio r of a geometric sequence are given. Find the sixth term and a formula for the nth term. $$a_{1}=10, r=-\frac{1}{2}$$

Step-by-Step Solution

Answer

Answer: The sixth term of the geometric sequence is -5/16, and the general formula for the nth term is \(a_n = 10*(-\frac{1}{2})^{n-1}\).

Step 1: Apply the Formula for the nth Term of a Geometric Sequence

To find \(a_6\), plug in the given values for \(a_1\), \(r\), and \(n\) into the formula: $$a_n = a_1*r^{n-1}$$ We are given that \(a_1 = 10\) and \(r = -\frac{1}{2}\). Since we want to find \(a_6\), we will substitute \(n = 6\) into the equation.

Step 2: Calculate \(a_6\)

Now we have: $$a_6 = 10*(-\frac{1}{2})^{6-1}$$ Simplify this expression: $$a_6 = 10*(-\frac{1}{2})^5 = 10*(-\frac{1}{32}) = -\frac{10}{32} = -\frac{5}{16}$$

Step 3: Generalize the Formula for \(a_n\)

Now that we have found \(a_6\), we can write a general formula for \(a_n\) by plugging in the given values for \(a_1\) and \(r\) back into the formula for the nth term of a geometric sequence: $$a_n = a_1*r^{n-1} = 10*(-\frac{1}{2})^{n-1}$$ So the sixth term of the geometric sequence is \(a_6 = -\frac{5}{16}\), and the general formula for the nth term is $$a_n = 10*(-\frac{1}{2})^{n-1}$$

Key Concepts

Understanding the nth Term Formula in Geometric Sequences

When dealing with geometric sequences, it is crucial to understand how to calculate any term in the sequence using a specific formula. This formula is:\[ a_n = a_1 \times r^{n-1} \]The formula is used to find the \( n^{th} \) term of a geometric sequence. Here's what each part of the formula represents:

• \( a_n \): The term you are trying to find in the sequence. For example, if you're looking for the sixth term, \( a_n \) would be \( a_6 \).
• \( a_1 \): This is the first term of the sequence. It is the starting point for the sequence.
• \( r \): The common ratio. It describes how the sequence progresses from one term to the next.
• \( n \): The term number you are targeting. If you need the sixth term, \( n \) will be 6.

In the example you provided, \( a_1 = 10 \) and \( r = -\frac{1}{2} \). To find the sixth term, substitute \( n = 6 \) into the formula.

The Role of the Common Ratio

In a geometric sequence, the common ratio \( r \) is what sets the sequence into motion. It acts as a multiplier that propels each term into the next. This ratio can be any real number, which includes fractions, whole numbers, or even negative numbers.

In the given exercise, the common ratio is \(-\frac{1}{2}\). This provides a critical clue about how each term relates to the previous term. With a common ratio of \(-\frac{1}{2}\):

• Each term is half the size of the previous term.
• Because it's negative, it flips the sign of each term in the sequence.

Knowing the common ratio allows you to fully understand how one term leads to the next and gradually build the sequence step by step.

Calculating Terms in a Geometric Sequence

Calculating terms in a geometric sequence becomes straightforward once you understand the nth term formula and the common ratio. Let's take it step by step with the exercise example.

You are tasked to find the sixth term. First, plug into the nth term formula: \[ a_6 = 10 \times \left(-\frac{1}{2}\right)^{6-1} \]Now solve the exponent:\(-\frac{1}{2}\) raised to the 5th power becomes:\((-\frac{1}{2})^5 = -\frac{1}{32}\)
Next, multiply by the first term:\[ 10 \times -\frac{1}{32} = -\frac{10}{32} \]Simplifying further gives you:\[ -\frac{5}{16} \]
And there you have it! The sixth term \( a_6 \) is \(-\frac{5}{16}\).
These steps demonstrate the sequence's behavior through successive multiplication by the common ratio. Understanding this process is crucial for working comfortably with any geometric sequence.