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

To find the common difference of the given arithmetic progression (AP), we start by understanding the general formula for the nth term of an AP. The nth term \(a_n\) of an AP can be expressed as:

\(a_n = a_1 + (n-1) \cdot d\)

where \(a_1\) is the first term, and \(d\) is the common difference.

For the given AP: \(\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2},\dots\), observe that:

• The first term \(a_1 = \sqrt{2}\).
• The second term \(a_2 = 2\sqrt{2}\).

The common difference \(d\) is calculated by subtracting the first term from the second term:

\(d = a_2 - a_1 = 2\sqrt{2} - \sqrt{2} = \sqrt{2}\)

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

Let's verify this with the next terms to ensure consistency:

• Third term \(a_3 = 3\sqrt{2}\)
• Fourth term \(a_4 = 4\sqrt{2}\)

The difference between consecutive terms:

• \(a_3 - a_2 = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}\)
• \(a_4 - a_3 = 4\sqrt{2} - 3\sqrt{2} = \sqrt{2}\)

Since the difference is the same for all terms, the common difference is confirmed as \(\sqrt{2}\).

Therefore, the correct option is \(\sqrt{2}\).