Problem 25 - Precalculus with Limits

Question

In Exercises 23 - 32, find a formula for for the arithmetic sequence. \( a_1 = 100, d = -8 \)

Step-by-Step Solution

Answer

The formula for the arithmetic sequence is \(a_n = 108 - 8n\).

Step 1: Identify the Given Values

In the problem, we are given the first term of the arithmetic sequence (\(a_1\)) to be 100. The common difference for this sequence (\(d\)) is -8.

Step 2: Use the Arithmetic Sequence Formula

With the given values, we now plug them into the arithmetic sequence formula, given as \(a_n = a_1 + (n - 1) * d\). So, substitute \(a_1 = 100\) and \(d = -8\) into the formula.

Step 3: Simplify

After substituting the given values into the arithmetic formula, the algebraic expression results in \(a_n = 100 - (n - 1)*8 \). We can simplify this further by distributing the -8 throughout the parentheses, resulting in \(a_n = 100 - 8n + 8\). And then add '100' and '8' together, gives the formula: \(a_n = 108 - 8n\).

Step 4: Final Answer

Once we have completed the simplification process, we are left with the final answer, which is \(a_n = 108 - 8n\). This is the formula for the given arithmetic sequence in the question.

Key Concepts

Common Difference

The common difference is a key element in understanding arithmetic sequences. It determines the constant amount added or subtracted as you progress from one term to the next. In an arithmetic sequence, every term after the first is developed by adding the common difference to the previous term. For understanding:

• If the common difference is positive, the sequence will increase.
• If it is negative, like in this problem where the common difference is -8, the sequence decreases.

Think of the common difference as the pattern or rule dictating how the sequence evolves. It precisely guides the direction and rate of change throughout the sequence. In the exercise given, since our common difference is -8, every successive term is 8 less than the preceding one, illustrating a clearly declining sequence.

First Term

The first term is the starting point of an arithmetic sequence, denoted by \(a_1\). It represents where the sequence begins and sets the stage for all subsequent calculations in developing the sequence. Understanding the first term is crucial because:

• It acts as the initial value that others build upon.
• All calculations of later terms are directly related to this foundational figure.

In our given exercise, the first term, \(a_1 = 100\), is the foundational block from which we calculate the rest of the sequence. If you view the sequence like a journey, \(a_1\) is the initial step and determines the starting path of our arithmetic sequence.

Sequence Formula

The sequence formula allows us to calculate any term in an arithmetic sequence without having to manually add the common difference repeatedly. It is typically expressed as:\[a_n = a_1 + (n - 1) \cdot d\]This formula gives you the flexibility to find the nth term directly and efficiently. Let's break down what each component means:

• \(a_n\) is the nth term you want to find.
• \(a_1\) is the first term, which serves as the baseline, or starting point.
• \(n\) represents the position of the term within the sequence (the ordinal number).
• \(d\) is the common difference, which dictates the step size from one term to the next.

By plugging in our known values (\(a_1 = 100\) and \(d = -8\)) into this formula, we derived: \(a_n = 108 - 8n\). This formula summarizes the problem elegantly, giving a direct method to calculate any term within our sequence.