Problem 17 - Algebra and Trigonometry

Question

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 3, \frac{3}{2}, 0,-\frac{3}{2}, \dots $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of \(-\frac{3}{2}\).

Step 1: Identify the Sequence Terms

The given sequence is: 3, \( \frac{3}{2} \), 0, \( -\frac{3}{2} \), ...

Step 2: Calculate the Differences Between Consecutive Terms

Find the difference between the consecutive terms to verify if it is constant. - Second term \( (\frac{3}{2}) - \) First term \((3) = \frac{3}{2} - \frac{6}{2} = -\frac{3}{2}\)- Third term \((0) - \) Second term \((\frac{3}{2}) = 0 - \frac{3}{2} = -\frac{3}{2}\)- Fourth term \( ( -\frac{3}{2}) - \) Third term \((0) = -\frac{3}{2} - 0 = -\frac{3}{2}\)

Step 3: Determine If the Sequence Is Arithmetic

Since the difference \(-\frac{3}{2}\) is the same for consecutive terms, the sequence has a common difference, indicating it is arithmetic.

Step 4: Find the Common Difference

The common difference is the value consistently obtained between consecutive terms of the sequence, which is \(-\frac{3}{2}\).

Key Concepts

Common Difference

The common difference in an arithmetic sequence is an essential concept that helps identify the pattern of the sequence. In simple terms, it is the amount added or subtracted from one term to the next. For an arithmetic sequence, this difference remains consistent throughout—a crucial characteristic that distinguishes it from other types of sequences.
To calculate the common difference, you subtract any term in the sequence from the term that follows it. For instance:

• Let's consider the sequence: 3, \( \frac{3}{2} \), 0, \(-\frac{3}{2} \), ...
• Subtract the first term (3) from the second term (\( \frac{3}{2} \)): \[\frac{3}{2} - 3 = -\frac{3}{2}\]
• Repeat this calculation for other consecutive terms, such as the difference between \( \frac{3}{2} \) and 0 or between 0 and \(-\frac{3}{2} \).
• Each time, the result is \(-\frac{3}{2}\).

When you observe a consistent difference like this between each pair of consecutive terms, you can confidently say that the common difference is \(-\frac{3}{2}\) and that the sequence is indeed arithmetic.

Sequence Terms

Understanding the terms of a sequence is the foundation of working with sequences. Each number in a sequence is called a term. In our specific case, the sequence provided is 3, \( \frac{3}{2} \), 0, \(-\frac{3}{2} \), and so on.
Terms in a sequence are usually written in a specific order, and they follow a particular rule or pattern. For arithmetic sequences, the terms are ordered such that each term increases or decreases by the same amount, known as the common difference. This pattern allows us to predict any term in the sequence given the starting term and the common difference.
To break it down:

• The first term of our sequence is 3.
• The second term is \( \frac{3}{2} \), obtained by subtracting the common difference \(-\frac{3}{2}\) from the first term.
• The third term is 0, which follows by subtracting the common difference from the second term further: \[ \frac{3}{2} - \frac{3}{2} = 0 \]
• The fourth term is \(-\frac{3}{2}\), continuing the established pattern.

Each term is therefore linked to its neighbors by this regular, predictable process.

Arithmetic Progression

Arithmetic progression is another name for arithmetic sequence, a simple yet powerful concept that showcases a linear pattern between numbers. This kind of progression is defined by the common difference, and knowing the first term and the common difference allows us to write the entire sequence.
The general formula for the \( n \)th term of an arithmetic progression is:\[ a_n = a_1 + (n-1) \cdot d \] where:

• \( a_n \) is the \( n \)th term of the sequence.
• \( a_1 \) is the first term (3 in our sequence).
• \( d \) is the common difference (\(-\frac{3}{2}\)).
• \( n \) is the term number in the sequence.

For example, to find the fourth term, plug in \( n = 4 \): \[ a_4 = 3 + (4-1) \cdot (-\frac{3}{2}) = 3 + 3 \cdot (-\frac{3}{2}) = 3 - \frac{9}{2} = -\frac{3}{2} \]
This confirms our calculated sequence. Arithmetic progression serves a multitude of real-world applications, from calculating simple interest in finance to determining patterns in data sets.