Problem 11 - Algebra and Trigonometry

Question

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{c_{n}\right\\}=\\{6-2 n\\} $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of -2. The first four terms are 4, 2, 0, and -2.

Step 1: Identify the general term

The general term for the sequence is given as \(c_{n} = 6 - 2n\). This will help verify if the sequence is arithmetic.

Step 2: Find the first four terms

Substitute the first four values of n into the general term \(c_{n} = 6 - 2n\). For n = 1: \(c_{1} = 6 - 2(1) = 4\)For n = 2: \(c_{2} = 6 - 2(2) = 2\)For n = 3: \(c_{3} = 6 - 2(3) = 0\)For n = 4: \(c_{4} = 6 - 2(4) = -2\)

Step 3: Check if the sequence is arithmetic

To determine if the sequence is arithmetic, find the difference between successive terms. \(c_{2} - c_{1} = 2 - 4 = -2\)\(c_{3} - c_{2} = 0 - 2 = -2\)\(c_{4} - c_{3} = -2 - 0 = -2\) Since the difference is constant, it confirms that the sequence is arithmetic.

Step 4: Identify the common difference

The common difference is the constant value found in Step 3. Therefore, the common difference d is -2.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is the difference between any two successive terms. It helps in identifying whether a sequence is arithmetic or not. In our given sequence with the general term \(c_{n} = 6 - 2n\), we calculated the first four terms: 4, 2, 0, and -2.

To find the common difference, we subtracted each term from the previous one.

• \(c_{2} - c_{1} = 2 - 4 = -2\)
• \(c_{3} - c_{2} = 0 - 2 = -2\)
• \(c_{4} - c_{3} = -2 - 0 = -2\)

This constant difference of -2 confirms that the given sequence is arithmetic. Understanding the common difference is crucial as it helps in forming any term in the sequence and verifying its nature.

General Term

The general term, also known as the nth term, of an arithmetic sequence provides a formula to calculate any term in the sequence without listing all previous terms. For our sequence, the general term is given by \(c_{n} = 6 - 2n\).

By substituting different values of n, we can determine the terms:

• For \(n = 1\), \(c_{1} = 6 - 2(1) = 4\)
• For \(n = 2\), \(c_{2} = 6 - 2(2) = 2\)
• For \(n = 3\), \(c_{3} = 6 - 2(3) = 0\)
• For \(n = 4\), \(c_{4} = 6 - 2(4) = -2\)

The general term simplifies the process of finding terms in an arithmetic sequence and is instrumental in understanding the structure of the sequence.

Sequence Verification

Verifying whether a given sequence is arithmetic involves checking if the difference between successive terms remains constant. Following our example, we derived the first four terms of the sequence using the general term \(c_{n} = 6 - 2n\): 4, 2, 0, and -2.

We then calculated the differences between successive terms:

• \(c_{2} - c_{1} = 2 - 4 = -2\)
• \(c_{3} - c_{2} = 0 - 2 = -2\)
• \(c_{4} - c_{3} = -2 - 0 = -2\)

Since each difference is -2, it confirms the sequence is arithmetic. Consistent common differences play a key role in sequence verification, ensuring that the sequence adheres to the properties of an arithmetic sequence.