Problem 64 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is an arithmetic sequence. \(f(n)=n^{2}-n+2\)

Step-by-Step Solution

Answer

No, \(f(n)\) is not an arithmetic sequence.

Step 1: Understand the Arithmetic Sequence Definition

An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is always constant. If \( f \) is an arithmetic sequence, then the difference \( f(n+1) - f(n) \) should be constant.

Step 2: Calculate General Term Difference

Find the expression for \( f(n+1) \).\[ f(n+1) = (n+1)^2 - (n+1) + 2 \]Now simplify this expression.\[ f(n+1) = n^2 + 2n + 1 - n - 1 + 2 = n^2 + n + 2 \]

Step 3: Find the Difference Between Consecutive Terms

Calculate \( f(n+1) - f(n) \).\[ f(n+1) - f(n) = (n^2 + n + 2) - (n^2 - n + 2) \]Simplify the expression.\[ f(n+1) - f(n) = n + n = 2n \]

Step 4: Determine if the Difference is Constant

The difference \( f(n+1) - f(n) = 2n \) is not constant because it depends on \( n \). For it to be an arithmetic sequence, this difference should not vary with \( n \).

Step 5: Conclusion

Since the difference between consecutive terms, \( 2n \), is not constant, \( f \) is not an arithmetic sequence.

Key Concepts

Understanding Sequence Definition in Arithmetic Sequences

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is always the same. This constant difference is known as the common difference. We often encounter arithmetic sequences in situations like evenly spaced numbers in a row or a consistent increase or decrease pattern.
To check if a sequence is arithmetic, we calculate the difference between consecutive terms. If this difference remains the same throughout the sequence, we can then confirm it as an arithmetic sequence.
In mathematical terms, if we have a sequence given by the formula \(a_n\), then for it to be arithmetic, the expression \(a_{n+1} - a_n\) must be a constant number. This means that each term in the sequence increases or decreases by the same amount.

Exploring Consecutive Terms

Consecutive terms in a sequence are those that come one after another in order. For example, in the sequence 5, 7, 9, 11, the numbers 5 and 7 are consecutive terms, as are the numbers 9 and 11.
When analyzing sequences, especially arithmetic ones, it's crucial to examine these consecutive terms to find the pattern or rule they follow.
For arithmetic sequences, we specifically compute the difference between each pair of consecutive terms. This difference will help us identify if the sequence has a constant step or not. If every consecutive pair of terms exhibits the same difference, the sequence is confirmed to be arithmetic.

Mastering Difference Calculation in Arithmetic Sequences

Calculating the difference between consecutive terms is essential to identify an arithmetic sequence. Let's take a sequence given as a function, like the one in the exercise: \(f(n) = n^2 - n + 2\). We aim to find \(f(n+1)\) and then \(f(n+1) - f(n)\) to see if it's constant.
First, substitute \(n+1\) into the sequence function:

• Calculate \( f(n+1) = (n+1)^2 - (n+1) + 2 = n^2 + 2n + 1 - n - 1 + 2 = n^2 + n + 2 \)

Next, find the difference \(f(n+1) - f(n)\):

• The difference is \((n^2 + n + 2) - (n^2 - n + 2) = n + n = 2n\)

Observe that \(2n\) is not a constant value, as it changes with different values of \(n\). Thus, the sequence \(f\) is not arithmetic because the difference between terms is not fixed.