Problem 23 - Contemporary Precalculus

Question

Show that the sequence is arithmetic and find its common difference. \(\\{c+2 n\\} \quad(c \text { constant })\)

Step-by-Step Solution

Answer

If so, what is the common difference? Answer: Yes, the sequence \({c+2n}\) is arithmetic with a common difference of \(2\).

Step 1: Find the first few terms of the sequence

First, we will find the first few terms of the sequence so that we can determine if there is a constant difference between consecutive terms. To do this, substitute the first few values of \(n\) (1, 2, 3, etc.) into the expression \(c+2n\): - For \(n=1\), term \(= c+2(1) = c+2\) - For \(n=2\), term \(= c+2(2) = c+4\) - For \(n=3\), term \(= c+2(3) = c+6 \) - For \(n=4\), term \(= c+2(4) = c+8\)

Step 2: Determine the common difference

Next, we will find the difference between consecutive terms to see if it is constant. - Difference between terms \(c+4\) and \(c+2\) is \((c+4) - (c+2) = 2\) - Difference between terms \(c+6\) and \(c+4\) is \((c+6) - (c+4) = 2\) - Difference between terms \(c+8\) and \(c+6\) is \((c+8) - (c+6) = 2\) Since the difference between consecutive terms is constant (2), we can conclude that the sequence is arithmetic.

Step 3: State the result

The sequence \({c+2n}\) is arithmetic with a common difference of \(2\).

Key Concepts

Common Difference

When it comes to understanding arithmetic sequences, the term common difference plays a crucial role. It refers to the consistent interval or difference between any two consecutive terms in such a sequence. For instance, if we have a list of numbers where each number is obtained by adding the same value to the previous number, that added value is called the common difference.

Using the exercise provided, when we calculate the difference between the terms \(c+4\) and \(c+2\), we subtract the latter from the former, giving us \(c+4 - (c+2) = 2\). We follow the same process for \(c+6\) and \(c+4\), as well as \(c+8\) and \(c+6\). The common difference in this case is consistently \(2\). This tells us that every term in the sequence is created by adding \(2\) to the previous term, rendering the sequence as arithmetic.

Consecutive Terms

When we speak of consecutive terms in an arithmetic sequence, we're referring to terms that come one after the other without any terms in between. A simple example would be the numbers \(4, 5, 6\), where \(4\) and \(5\), as well as \(5\) and \(6\), are consecutive.

In the context of the exercise, when we analyze terms like \(c+2\), \(c+4\), \(c+6\), and \(c+8\), they represent consecutive terms where the value of \(n\) increases by \(1\) each time—leading from one term to the next. Checking the difference between consecutive terms gives us insight into whether the sequence follows a specific, predictable pattern—one of the hallmarks of being an arithmetic sequence.

Arithmetic Progression

An arithmetic progression, also known as an arithmetic sequence, is a series of numbers in which the difference between any two consecutive terms is constant. This defining characteristic means that each term can be formulated by adding a particular value, the common difference, to the preceding term.

Take the sequence from our exercise, \(c+2n\). Once we establish that the sequence has a common difference (which we did by calculating the differences between its consecutive terms to be \(2\)), we can confidently label it as an arithmetic sequence. It’s a straightforward pattern that allows for easy prediction of subsequent terms and lays the foundation for various applications in mathematics, including solving problems on sums of sequences and series.