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:

To avoid calculation errors with irrational terms, think of $\sqrt{2}$ as an algebraic variable, say $x$.
The sequence then becomes $x, 2x, 3x, 4x, \dots$
The common difference is simply $2x - x = x$, which is $\sqrt{2}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks us to find the common difference of the given Arithmetic Progression (AP).
An Arithmetic Progression is a sequence of numbers where the difference between consecutive terms remains constant throughout the sequence.

Step 2: Key Formula or Approach:
The common difference, denoted by $d$, is calculated as the difference between any term and its preceding term:
\[ d = a_{n+1} - a_n \]
For the first few terms, this can be written as:
\[ d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 \]

Step 3: Detailed Explanation:

• Identify the individual terms of the given progression:
- First term, $a_1 = \sqrt{2}$
- Second term, $a_2 = 2\sqrt{2}$
- Third term, $a_3 = 3\sqrt{2}$
- Fourth term, $a_4 = 4\sqrt{2}$

• Calculate the difference between the second term and the first term:
\[ d = a_2 - a_1 \]
\[ d = 2\sqrt{2} - \sqrt{2} \]
Factoring out the radical term $\sqrt{2}$:
\[ d = (2 - 1)\sqrt{2} = 1\sqrt{2} = \sqrt{2} \]

• Verify the calculation using the third and second terms:
\[ d = a_3 - a_2 \]
\[ d = 3\sqrt{2} - 2\sqrt{2} \]
\[ d = (3 - 2)\sqrt{2} = \sqrt{2} \]

• Verify again using the fourth and third terms:
\[ d = a_4 - a_3 \]
\[ d = 4\sqrt{2} - 3\sqrt{2} = \sqrt{2} \]

• Since the difference between consecutive terms is consistently $\sqrt{2}$, this value is the common difference of the Arithmetic Progression.


Step 4: Final Answer:
The common difference of the given AP is $\sqrt{2}$.
Hence, option (A) is correct.