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

To find the common difference of the given arithmetic progression (AP), we first need to recall the definition of an AP. An arithmetic progression is a sequence of numbers such that the difference between any two successive terms is constant. This constant difference is known as the "common difference."

The given AP is: \(\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \dots\)

1. Let's identify the first few terms of the sequence:
   • First term, \(a_1 = \sqrt{2}\)
   • Second term, \(a_2 = 2\sqrt{2}\)
   • Third term, \(a_3 = 3\sqrt{2}\)
   • Fourth term, \(a_4 = 4\sqrt{2}\)
2. To find the common difference \(d\), we subtract the first term from the second term:
   • \(d = \sqrt{2} (2 - 1) = \sqrt{2}\)
3. You can verify the common difference by checking the difference between subsequent terms:
   • \(d = a_3 - a_2 = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}\)
   • \(d = a_4 - a_3 = 4\sqrt{2} - 3\sqrt{2} = \sqrt{2}\)

Thus, the common difference of the given arithmetic progression is \(\sqrt{2}\).

Conclusion: The correct option for the common difference is \(\sqrt{2}\).