Problem 23 - Algebra and Trigonometry

Question

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 0.0004,-0.004,0.04,-0.4, \ldots $$

Step-by-Step Solution

Answer

The formula for the general term of the sequence is \(a_{n} = 0.0004 \cdot (-10)^{(n-1)} \) and the seventh term of the sequence is -4000.

Step 1: Identify the first term and common ratio

The first term \(a = 0.0004\). The common ratio \(r\) can be determined by dividing the second term by the first term, i.e. \(r = -0.004 / 0.0004 = -10\)

Step 2: Formulate the General Formula

The general formula for the nth term of a geometric series is given by \(a_{n} = a \cdot r^{(n-1)} \). Substituting the identified first term and common ratio, the formula becomes \(a_{n} = 0.0004 \cdot (-10)^{(n-1)} \)

Step 3: Find the Seventh Term

Substitute \(n=7\) into the formula to find the seventh term: \(a_{7} = 0.0004 \cdot (-10)^{(7-1)} = -4000 \)

Key Concepts

nth term formula

The nth term formula is key to understanding geometric sequences. A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Here's the basic idea: if you know the formula for the nth term, you can find any term in the sequence without having to list all the preceding terms.

The nth term formula for a geometric sequence is given by:

• \( a_n = a \cdot r^{(n-1)} \)

In this formula:

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

To use this formula effectively, you first need to identify the first term and the common ratio. Then, simply plug these values into the formula along with the desired term number \( n \). This makes calculating any term in the sequence simple and straightforward.

common ratio

The common ratio in a geometric sequence is the factor by which we multiply to get from one term to the next. It's consistent across the sequence, making it predictable and easy to work with. Knowing the common ratio allows you to use the nth term formula effectively. To find the common ratio \( r \), divide any term in the sequence by the preceding term. Let's understand it further:

In the sequence given in the exercise:

• The first two terms are 0.0004 and -0.004.
• To find the common ratio, divide the second term by the first: \( r = \frac{-0.004}{0.0004} = -10 \).

The result \(-10\) is the common ratio for this sequence. This negative value indicates that the sequence's terms will alternate in sign as you progress through it. Understanding the common ratio helps in constructing the general formula and predicting future terms.

seventh term

Finding specific terms in a geometric sequence, such as the seventh term, is straightforward once you have the nth term formula. With the formula ready, you just need to substitute the term number and compute the result.

According to the solution provided, the general formula for the nth term for this sequence is:

• \( a_n = 0.0004 \cdot (-10)^{(n-1)} \)

To find the seventh term \( a_7 \), plug in \( n = 7 \) into the formula:

• \( a_7 = 0.0004 \cdot (-10)^{6} \)

Calculating this gives:

• \( a_7 = 0.0004 \cdot 1000000 = -4000 \)

This calculation shows that the seventh term in the sequence is \(-4000\). The negative sign here reaffirms the alternating pattern in this specific sequence. By following these steps, you can easily find any other term in the sequence as well!