Question

For the Arithmetic Progression : \(\frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}, \ldots\) the common difference \(d\) is :

• A. -1
• B. -2
• C. -3
• D. 1

Hint:

Subtract the first term from the second: \(1/2 - 3/2\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
In an arithmetic progression each term is got by adding the same number to the previous term. That number is the common difference \(d\).

Step 2: Key Formula or Approach:
\[ d = a_{2} - a_{1} \]
Any two neighbouring terms will do.

Step 3: Detailed Explanation:
\[ d = \frac{1}{2} - \frac{3}{2} = -\frac{2}{2} = -1 \]
Check with the next pair: \(-\dfrac{1}{2} - \dfrac{1}{2} = -1\). And \(-\dfrac{3}{2} - \left(-\dfrac{1}{2}\right) = -1\). All gaps are the same.

Step 4: Check the other options:
-2 and -3 are too large a drop each time. +1 has the wrong sign, because the terms are going down. So only -1 works.

Final Answer:
The common difference is -1. \[ \boxed{d = -1} \]