Question

Which of the following sequence is not an A.P. ?

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

Hint:

In option (C), simplify radical terms like \( \sqrt{8} = 2\sqrt{2} \) to see the common difference more easily.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A sequence is an Arithmetic Progression (A.P.) if the difference between consecutive terms is constant.
Step 2: Detailed Explanation:
Check differences for each option:
(A) \( d = \frac{5}{2} - 2 = 0.5 \); \( 3 - \frac{5}{2} = 0.5 \). Constant \( d = 0.5 \). (It is an A.P.)
(B) \( d = -3.2 - (-1.2) = -2 \); \( -5.2 - (-3.2) = -2 \). Constant \( d = -2 \). (It is an A.P.)
(C) \(\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, \dots \) differences are \( \sqrt{2}, \sqrt{2}, \dots \) (It is an A.P.)
(D) \( 1, 9, 25, 49, \dots \)
\( 9 - 1 = 8 \)
\( 25 - 9 = 16 \)
The differences are not constant (\( 8 \neq 16 \)).
Step 3: Final Answer:
The sequence \( 1^2, 3^2, 5^2, 7^2, \dots \) is not an A.P.