Question

Which of the following sequence is \(\textit{not }\)an A.P. ?
 

• A. \( 2, \frac{5}{2}, 3, \frac{7}{2}, \dots \)
• B. \( -1.2, -3.2, -5.2, -7.2, \dots \)
• C. \( \sqrt{2}, \sqrt{8}, \sqrt{18}, \dots \)
• D. \( 1^2, 3^2, 5^2, 7^2, \dots \)

Hint:

Always simplify radical terms like \( \sqrt{8} \) to \( 2\sqrt{2} \) before checking the common difference.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
An Arithmetic Progression (A.P.) is a sequence where the difference between consecutive terms is constant.
Step 2: Detailed Explanation:
Let's check the common difference \( d \) for each:
(A) \( 2.5 - 2 = 0.5 \); \( 3 - 2.5 = 0.5 \). It is an A.P.
(B) \( -3.2 - (-1.2) = -2 \); \( -5.2 - (-3.2) = -2 \). It is an A.P.
(C) \( \sqrt{2}, 2\sqrt{2}, 3\sqrt{2} \dots \). Difference is \( \sqrt{2} \). It is an A.P.
(D) \( 1, 9, 25, 49 \). Differences are: \( 9 - 1 = 8 \) and \( 25 - 9 = 16 \).
Since \( 8 \neq 16 \), the common difference is not constant.
Step 3: Final Answer:
The sequence \( 1^2, 3^2, 5^2, 7^2, \dots \) is not an A.P.