Problem 67 - College Algebra

Question

Consider the sequence defined by \(a_{n}=-6-8 n .\) Is \(a_{n}=-421\) a term in the sequence? Verify the result.

Step-by-Step Solution

Answer

No, -421 is not a term in the sequence.

Step 1: Understanding the Problem

We need to determine if there exists a natural number \(n\) such that \(a_n = -421\) in the sequence defined by \(a_n = -6 - 8n\). If such an \(n\) exists, \(-421\) is a term in the sequence.

Step 2: Setting Up the Equation

Substitute \(a_n = -421\) into the sequence formula \(a_n = -6 - 8n\) to set up the equation: \(-6 - 8n = -421\).

Step 3: Solving for n

Rearrange the equation to solve for \(n\):1. Add 6 to both sides: \[-8n = -421 + 6\]2. Simplify the right side: \[-8n = -415\]3. Divide both sides by -8: \[n = \frac{-415}{-8}\]4. Simplify the division: \[n = 51.875\]

Step 4: Evaluating the Result

Since \(n = 51.875\) is not a whole number, \(-421\) cannot be a term in the sequence, as \(n\) must be a natural number (whole number greater than zero).

Key Concepts

Solving Equations

In mathematics, solving equations is like finding the "balance point." When you have an equation, think of it like a scale—you want both sides to weigh the same. In our example, the equation was \(-6 - 8n = -421\). Here’s how we solved it: we wanted to isolate \(n\), so all that was left was \(n\) on one side. First, by adding 6 to both sides, we kept the scale balanced but moved closer to isolating \(n\). The equation became \(-8n = -415\). Next, we divided both sides by \(-8\) to completely isolate \(n\), leading us to check for a solution that tells us about the sequence conditions. When solving equations, remember:

• Aim to keep operations balanced: whatever you do to one side, do to the other too.
• Choose operations that simplify the equation. Here it was addition and division.

This method helps pinpoint the exact value of the unknown variable, which in this case helps us understand the sequence.

Sequence Formula

A sequence formula is like a recipe that describes each term in a sequence based on its position. The sequence given by \(a_{n} = -6 - 8n\) tells us how each term relates to \(n\), the position in the sequence. In this scenario:

• \(-6\) is the starting point or "offset" of the sequence. It’s the initial value before adjustments.
• \(-8n\) describes the change between terms relative to their position \(n\).

By plugging different values of \(n\) into \(a_{n} = -6 - 8n\), you can find out what each term is. However, only whole numbers for \(n\) make sense if we're looking for actual positions in the sequence, because you can't have a half position in a list. Understanding the structure of the formula helps predict and calculate any term throughout the sequence efficiently.

Natural Numbers

Natural numbers are the building blocks of our number system. They start from 1 and go up infinitely. Think of them like counting numbers: 1, 2, 3, and so on. They don’t include zero, negatives, or fractions. In the context of sequences, the position \(n\) is considered a natural number because it indicates order in a list. Since a sequence has a clear starting point and progresses in discrete steps, using natural numbers fits this scenario perfectly. For example, in determining if \(a_{n} = -421\) is part of the sequence, \(n\) should be a natural number. That's why discovering \(n = 51.875\) told us that \(-421\) isn't a valid term— the sequence only accommodates terms with whole, positive \(n\).Understanding this core concept ensures you’re looking at sequences with the right lens – one that matches real-world counting and organizing.