Problem 22 - Contemporary Precalculus

Question

Show that the sequence is arithmetic and find its common difference. $$\left\\{\frac{\pi-n}{2}\right\\}$$

Step-by-Step Solution

Answer

If so, what is the common difference? Answer: Yes, the given sequence is arithmetic with a common difference of -1.

Step 1: Find the general term of the sequence

First, let's write down the general term of the given sequence: $$a_n = \frac{\pi-n}{2}$$

Step 2: Find the difference between two consecutive terms in the sequence

To check if the sequence is arithmetic, we need to find the difference between consecutive terms: $$d = a_{n+1} - a_n$$ Let's substitute the expression for the general term, \(a_n\): $$d = \left(\frac{\pi-(n+1)}{2}\right) - \left(\frac{\pi-n}{2}\right)$$

Step 3: Simplify the difference between consecutive terms

Now, let's simplify the expression for the difference between consecutive terms: $$d = \frac{\pi-(n+1)}{2} - \frac{\pi-n}{2}$$ $$= \frac{\pi-n-1}{2} - \frac{\pi-n}{2}$$ $$= \frac{(-1)(2)}{2}$$ $$= -1$$

Step 4: Determine if the sequence is arithmetic and find the common difference

Since the difference between any two consecutive terms is constant, the sequence is arithmetic. In our case, the common difference \(d\) is equal to -1. Therefore, the given sequence is an arithmetic sequence with a common difference of -1.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is the consistent difference between two consecutive terms. Imagine stepping up a staircase, each step being a fixed height above the last; that's the common difference.
In the context of arithmetic sequences, if you find an unchanging jump in numbers, you're looking at the common difference. It is often represented by the letter 'd'.
For any arithmetic sequence, to find this common difference, you simply subtract one term from the term that follows it. For instance, in our example sequence, we calculated:

• \(d = a_{n+1} - a_n\)
• Which became \[d = \left(\frac{\pi-(n+1)}{2}\right) - \left(\frac{\pi-n}{2}\right)\] leading to finally obtaining \(-1\) as our common difference.

This predictable behavior is what makes arithmetic sequences straightforward and easy to deal with. You can always rest assured that each term is merely the previous term increased or decreased by this constant value.

Consecutive Terms

Consecutive terms in a sequence are like neighbors in a row. They follow one after another without any skip in between. When dealing with arithmetic sequences, each term is derived straightforwardly from the one before it.
Consecutive terms can be represented mathematically as \(a_n, a_{n+1}, a_{n+2}, \ldots\), where each succeeding term can be expressed based on its predecessor.
In an arithmetic sequence, understanding consecutive terms is crucial: they help in spotting the constant common difference. Consider our sequence, \(\frac{\pi-n}{2}\), where:

• \(a_n = \frac{\pi-n}{2}\)
• \(a_{n+1} = \frac{\pi-(n+1)}{2}\)

As seen, each term is derived directly from its predecessor by a modification of \(-1\). This consistency in change is what makes it easy to identify and understand arithmetic sequences.

General Term

The general term of an arithmetic sequence is a formula that allows you to find any term in that sequence without needing to write all the previous terms. It acts like a direct shortcut, saving you time and effort, especially for large sequences.
For any arithmetic sequence, the general term is typically denoted as \(a_n\) and can be formulated as:

• \(a_n = a_1 + (n-1) \cdot d\)

where \(a_1\) is the first term and \(d\) is the common difference.
In our example, the sequence \(a_n = \frac{\pi-n}{2}\) was directly given as the general term form. Here,

• \(a_n = \frac{\pi-n}{2}\) for \(n = 1, 2, 3, \ldots\)

This means every term in the sequence can be calculated using this single formula without repetitive calculations. This feature of the general term underscores its power, helping you predict any part of the sequence efficiently.