Problem 63 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is an arithmetic sequence. $$f(n)=4 n-(3-n)$$

Step-by-Step Solution

Answer

Yes, \(f(n) = 5n - 3\) is an arithmetic sequence with a common difference of 5.

Step 1: Simplify the Expression

We start by simplifying the given formula for the function \(f(n) = 4n - (3 - n)\). Distribute the subtraction across the parentheses: \(4n - 3 + n\). This simplifies further to \(5n - 3\).

Step 2: Identify the Sequence Structure

We know an arithmetic sequence has the form \(a_n = a_1 + (n-1)d\), where \(d\) is a constant common difference. Here, the formula for \(f(n)\) after simplification resembles \(5n - 3\), which can be written as \(f(n) = 5n + (-3)\).

Step 3: Verify the Common Difference

The general term of an arithmetic sequence can be expressed as \(a_n = a + (n-1)d\). Comparing this with \(f(n) = 5n - 3\), the term is linear with respect to \(n\). The coefficient of \(n\), which is \(5\), represents the common difference \(d\). Here, \(d = 5\), indicating it is indeed an arithmetic sequence.

Key Concepts

Common Difference

When discussing arithmetic sequences, one of the key elements is the common difference. This is the value by which each term in the sequence increases or decreases as you progress through the sequence. An arithmetic sequence can be described using the formula:\[a_n = a_1 + (n-1)d\]where:

• \(a_n\) is the nth term of the sequence
• \(a_1\) is the first term
• \(d\) is the common difference

In our exercise, we have simplified the expression for \(f(n)\) to be \(5n - 3\). The coefficient of \(n\) is \(5\), which represents the common difference \(d\). So, every step you go forward in your sequence, you’re adding \(5\) to the previous term. This constant addition confirms that the sequence is arithmetic.

Sequence Structure

The structure of an arithmetic sequence is linear, which means the terms follow a straight-line pattern on a graph. Each term increases by the same fixed amount, known as the common difference, as you move from one term to the next. This structure can be expressed by a simple linear equation.From our simplified function, \(f(n) = 5n - 3\), we can see it matches the form \(a_n = a_1 + (n-1)d\). The function describes a line with a consistent slope given by the common difference. In this specific scenario:

• The expression \(5n\) suggests each term increases linearly as \(n\) increases.
• The \'-3\' can be considered as a shift downwards which affects the starting point of the sequence.

This overall structure confirms that the sequence is arithmetic, with the terms regularly following the linear pattern described by the common difference.

Simplifying Expressions

Simplifying expressions is crucial for identifying the characteristics of a sequence. It often involves combining like terms and eliminating parentheses to reveal a clearer form. For the function \(f(n) = 4n - (3 - n)\), simplifying is a necessary step.By distributing and combining like terms:

• Distribute the \(-\) sign into the parentheses: \(4n - 3 + n\).
• Then combine the like terms \(4n\) and \(n\) to get \(5n\).
• We end up with \(5n - 3\).

The simplified form \(5n - 3\) presents a much clearer view of the arithmetic sequence structure, where \(5\) is the common difference and \(-3\) is a constant term. Simplifying expressions helps in quickly identifying patterns and constants, making it easier to determine if a sequence is arithmetic or has other mathematical properties.