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:

When working with surds (roots), treat the radical like a variable. For example, \(2\sqrt{2} - \sqrt{2}\) is just like \(2x - x\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (\(d\)).
Step 2: Key Formula or Approach:
The common difference is calculated as: \[ d = a_{n} - a_{n-1} \] Commonly, \(d = a_2 - a_1\).
Step 3: Detailed Explanation:
1. Let the first term \(a_1 = \sqrt{2}\). 2. Let the second term \(a_2 = 2\sqrt{2}\). 3. Calculate the difference: \[ d = 2\sqrt{2} - \sqrt{2} \] \[ d = \sqrt{2}(2 - 1) = \sqrt{2} \] 4. Verification: \(3\sqrt{2} - 2\sqrt{2} = \sqrt{2}\). The difference is consistent.
Step 4: Final Answer:
The common difference is \(\sqrt{2}\).