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:

Think of terms like $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}$ as $1x, 2x, 3x$ where $x = \sqrt{2}$.
The step-to-step difference is simply $x$, which is $\sqrt{2}$.
This simplification makes calculations with surds much easier to visualize mentally.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic is Arithmetic Progressions (APs).
We are given a sequence of numbers and asked to determine its common difference.

Step 2: Key Formula or Approach:
An Arithmetic Progression is a sequence of numbers in which the difference between any two consecutive terms is constant.
This constant difference is called the common difference ($d$).
The formula to calculate the common difference is:
\[ d = a_{n+1} - a_n \]
For example, we can calculate $d$ using the first and second terms:
\[ d = a_2 - a_1 \]

Step 3: Detailed Explanation:

• Identify the given sequence of terms:
First term ($a_1$) = $\sqrt{2}$
Second term ($a_2$) = $2\sqrt{2}$
Third term ($a_3$) = $3\sqrt{2}$
Fourth term ($a_4$) = $4\sqrt{2}$

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

• Verify this difference with the next consecutive terms to ensure consistency:
\[ a_3 - a_2 = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}(3 - 2) = \sqrt{2} \]
\[ a_4 - a_3 = 4\sqrt{2} - 3\sqrt{2} = \sqrt{2}(4 - 3) = \sqrt{2} \]

• Since the difference is constant at each step, the sequence is indeed an AP, and the common difference is $\sqrt{2}$.


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