Question

The common difference of the AP : \(\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \dots\) is :

• A. \(\sqrt{2}\)
• B. 1
• C. \(2\sqrt{2}\)
• D. \(-\sqrt{2}\)

Hint:

Treat radicals like variables. Just as \(2x - x = x\), \(2\sqrt{2} - \sqrt{2} = \sqrt{2}\). Always verify with the third term: \(3\sqrt{2} - 2\sqrt{2} = \sqrt{2}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In an Arithmetic Progression (AP), the common difference (\(d\)) is the constant value obtained by subtracting any term from its succeeding term.
Step 2: Key Formula or Approach:
Common difference \(d = a_{2} - a_{1} = a_{3} - a_{2}\).
Step 3: Detailed Explanation:
Given AP: \(\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \dots\)
Here, first term \(a_{1} = \sqrt{2}\) and second term \(a_{2} = 2\sqrt{2}\).
\[ d = a_{2} - a_{1} \]
\[ d = 2\sqrt{2} - \sqrt{2} \]
\[ d = \sqrt{2}(2 - 1) \]
\[ d = \sqrt{2} \]
Step 4: Final Answer:
The common difference of the given AP is \(\sqrt{2}\).