Problem 14 - A Graphical Approach to College Algebra

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=-3, d=-4$$

Step-by-Step Solution

Answer

\(a_8 = -31\), \(a_n = -4n + 1\).

Step 1: Understand the Problem

We need to find the 8th term of the sequence, denoted as \(a_8\), and the general term \(a_n\) for an arithmetic sequence given the first term \(a_1 = -3\) and the common difference \(d = -4\).

Step 2: Formula for the General Term

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1) imes d\). We will use this formula to find both \(a_8\) and the expression for \(a_n\).

Step 3: Substitute to Find \(a_8\)

Using the formula \(a_n = a_1 + (n-1) imes d\), substitute \(n = 8\), \(a_1 = -3\), and \(d = -4\) to find \(a_8\):\[a_8 = -3 + (8-1) imes (-4)\]

Step 4: Calculate \(a_8\)

Compute the expression:\[a_8 = -3 + 7 imes (-4) = -3 - 28 = -31\]Thus, the 8th term \(a_8\) is \(-31\).

Step 5: Derive the General Term Formula for \(a_n\)

Using the general term formula again, we continue with \(a_1 = -3\) and \(d = -4\) to derive \(a_n\): \[a_n = -3 + (n-1) imes (-4)\]Simplify the expression:\[a_n = -3 - 4(n-1) = -3 - 4n + 4\]Combine like terms:\[a_n = -4n + 1\]So, the general term \(a_n = -4n + 1\).

Key Concepts

General Term Formula

An arithmetic sequence likes to keep things simple. It builds itself steadily based on a specific pattern, starting from the first term and adding a constant amount each time to get to the next term. The secret to uncovering any term in this sequence lies in the General Term Formula. It's like a magic key.

The formula is expressed as \( a_n = a_1 + (n-1) \times d \). Here’s what each part means:

• \( a_n \) is the term you're trying to find.
• \( a_1 \) is the very first term of the sequence.
• \( d \) is the common difference, showing how much you add each time.
• \( n \) is the term number in the sequence.

By plugging the right numbers into this formula, you can find any specific term in the sequence, like \( a_8 = a_1 + (8-1) \times d \), to get the eighth term.

Common Difference

The common difference in an arithmetic sequence is a simple yet powerful tool. This number tells you exactly how much to add or subtract as you move from one term to the next. It guides the entire sequence like a faithful guide.

In the discussed exercise, the common difference is \( d = -4 \). That negative sign tells us that this sequence decreases by 4 each step. To find this for any sequence:

• Pick any consecutive terms in the sequence.
• Subtract the first term from the second.

This value will remain the same for all steps in the sequence because of its regularity and predictability.

Arithmetic Progression

Arithmetic Progression is another name for the easy-going series of numbers we are working with. It's a sequence where each term after the first is formed by adding the common difference to the previous term.

Imagine starting with \( a_1 = -3 \). Add the common difference of \( d = -4 \) each time, and you simply walk slowly down the number line.

• First Term: \( a_1 = -3 \)
• Second Term: \( a_2 = -3 + (-4) = -7 \)
• Third Term: \( a_3 = -7 + (-4) = -11 \)
• And so on...

Each next term emerges from this straightforward pattern, showing the power of arithmetic progression.

Sequence Calculation

Calculating terms in an arithmetic sequence is like following a simple recipe – it has steps to repeat, leading to a clear instruction. To calculate terms, use the General Term Formula. For example, to find \( a_8 \):

• First, identify the given values: \( a_1 = -3 \), \( d = -4 \), and \( n = 8 \).
• Then, insert these into the formula: \( a_8 = -3 + (8-1) \times (-4) \).
• Solve step by step: \( a_8 = -3 + 7 \times (-4) \).
• Complete the multiplication: \( 7 \times (-4) = -28 \).
• Finally, add everything together: \( -3 - 28 = -31 \).

And there you have it, \( a_8 = -31 \) easily thanks to the formula and those arithmetic directions.