Problem 67 - Algebra 2

Question

The first term of a geometric sequence is \(-1 .\) The common ratio is \(-5 .\) Find the eighth term in the sequence.

Step-by-Step Solution

Answer

The eighth term in the sequence is 3125.

Step 1: Identify the Values

From the problem, we identify the first term (a) as -1, the common ratio (r) as -5, and the term to find (n) is the 8th term.

Step 2: Substitute into the Formula

The nth term of a geometric sequence is calculated by: \(a*r^{(n-1)}\). Substituting the known values gives: \(-1*(-5)^{8-1}\)

Step 3: Solve to Find the 8th Term

By simplifying, we get \(-1*(-5)^7\), which equals 3125.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a key factor. It is the constant value that each term in the sequence is multiplied by to get the next term. This means if the common ratio is negative, like in our original exercise, the sequence will alternate signs as it progresses.
If we start with the first term of a sequence, we can find the next terms by repeatedly multiplying by the common ratio. For example, if the first term is -1 and the common ratio is -5, the second term would be \(-1 \times (-5) = 5\).

• The common ratio can be any real number.
• If it’s greater than 1, terms get larger.
• If it’s between 0 and 1, terms get smaller.
• An absolute value of 1 means all terms are the same.

Understanding the common ratio helps us predict how a sequence behaves and lets us easily calculate any term.

Finding the Nth Term

The 'nth term' simply refers to any term in a geometric sequence whose position we want to find. To determine the nth term, we often use specific values like the first term and the common ratio, as shown in the original problem.
To find the nth term, we use the formula for the nth term of a geometric sequence:

• Identify the first term, \(a\).
• Identify the common ratio, \(r\).
• Determine which term number, \(n\), you’re solving for.

Using these elements, we can plug into the formula \(a \cdot r^{(n-1)}\) to calculate any desired nth term of a sequence. In the context of the example, we found the eighth term or \(n = 8\), which was calculated as \(-1 \cdot (-5)^{7}\).
This gives us a powerful tool for exploring sequences without needing to write out every single term.

Exploring the Geometric Sequence Formula

The geometric sequence formula is a concise expression that describes how to find any term in a geometric sequence. It allows you to skip directly to any term without having to calculate every term before it. The formula is:\[ a_n = a \cdot r^{(n-1)} \]Where:

• \(a_n\) is the nth term we're trying to find.
• \(a\) is the first term in the sequence.
• \(r\) is the common ratio of the sequence.
• \(n\) is the position of the term in the sequence.

This formula streamlines finding terms and is essential for solving problems quickly. You simply insert the known values, as was demonstrated by finding the eighth term in the given exercise, simplifying the process to identifying terms efficiently.