Problem 24 - College Algebra

Question

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. $$0.0007,-0.007,0.07,-0.7, \ldots$$

Step-by-Step Solution

Answer

The formula for the nth term of the sequence is given by \(a_{n} = 0.0007 * (-10)^{n-1}\). The seventh term of the sequence is 700.

Step 1: Identify the first term (a_{1}) and the common ratio (r)

First, we identify the first term and the common ratio of the sequence. The first term \(a_{1}\) is 0.0007. The common ratio r can be found by dividing any term by the term directly before it. Here, -0.007/0.0007 gives -10, which tells us that the common ratio is -10.

Step 2: Write down the general formula for the nth term

Now we insert \(a_{1}\) = 0.0007 and r = -10 into the general formula for a geometric sequence, which gives: \(a_{n} = 0.0007 * (-10)^{n-1}\). This formula can be used to find any term in this geometric sequence.

Step 3: Find the seventh term

Now we use the formula we found to find the seventh term of the sequence. We plug n = 7 into the formula to get: \(a_{7} = 0.0007 * (-10)^{7-1} = 0.0007 * (-10)^6 = 700

Key Concepts

Nth Term of a Geometric Sequence

Understanding how to find the nth term of a geometric sequence is essential for working with patterns in numbers. Imagine you're given a series of numbers which multiply by the same amount each time; that's a geometric sequence. To identify any term in this ordered list, especially if it's far down the line, you would use the nth term formula:
\( a_n = a_1 \times r^{(n-1)} \)
Here, \( a_n \) is the nth term you're seeking, \( a_1 \) is the first term in the sequence, \( r \) is the common ratio (the amount each term is multiplied by to get the next term), and \( n \) signifies the term's position in the sequence. For instance, if you wanted to determine the 7th term of a sequence starting with 0.0007 that multiplies by -10 each time, you would plug those values into the formula to find \( a_7 \).

Common Ratio

The common ratio in a geometric sequence is the consistent factor you multiply (or divide) one term by to get the next term. It is a central concept and can be denoted as \( r \). You can find the common ratio by dividing any term in the sequence by the previous term. For example, if you have a sequence where the second term is -0.007 and the first term is 0.0007, divide -0.007 by 0.0007 to get -10. This means each term is 10 times the previous term but with an alternating sign, which indicates a negative common ratio. Identifying the correct common ratio is crucial for the formula of the nth term to work.

Geometric Progression

A geometric progression, also known as a geometric sequence, is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This progression creates a pattern that can expand infinitely or until a certain term.
Geometric progressions are everywhere—in finance for calculating compound interest, in biology for calculating population growth, or in physics for understanding aspects of waves. Recognizing and working with these patterns allows for predictions of future terms and deeper insights into the nature of the sequence.

Algebraic Expressions

In algebra, expressions are combinations of symbols and numbers that represent a mathematical relationship. The expression for the nth term of a geometric sequence is a perfect example of how algebra can describe numerical patterns.
An algebraic expression for the nth term includes variables (such as \( n \) for term number) and constants (like the first term and the common ratio). It doesn't solve for a number but sets up a relationship that allows you to find specific terms within a sequence when certain values are known. Mastering algebraic expressions is a fundamental skill that helps in resolving various mathematical and real-world problems by providing a formulaic approach to finding solutions.