Question

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

• A. 1, 2, 3, 4, ...
• B. 3, 6, 9, 12, ...
• C. 2, 4, 6, 8, ...
• D. \(2^2, 4^2, 6^2, 8^2, ...\)

Hint:

Be careful with sequences involving squares or powers. They are rarely Arithmetic Progressions. Always write out the first few terms of the sequence to make the pattern clear before calculating differences.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
An Arithmetic Progression (A.P.) is a sequence of numbers where the difference between consecutive terms is constant. We need to find the sequence that does not have a constant difference.

Step 2: Key Formula or Approach:
We will check the difference between consecutive terms for each sequence. If the difference is not constant, the sequence is not an A.P.

Step 3: Detailed Explanation:
Let's analyze each option:
(A) 1, 2, 3, 4, ...
\(2 - 1 = 1\), \(3 - 2 = 1\), \(4 - 3 = 1\). The common difference is 1. This is an A.P.
(B) 3, 6, 9, 12, ...
\(6 - 3 = 3\), \(9 - 6 = 3\), \(12 - 9 = 3\). The common difference is 3. This is an A.P.
(C) 2, 4, 6, 8, ...
\(4 - 2 = 2\), \(6 - 4 = 2\), \(8 - 6 = 2\). The common difference is 2. This is an A.P.
(D) \(2^2, 4^2, 6^2, 8^2, ...\) which is the sequence 4, 16, 36, 64, ...
\(16 - 4 = 12\)
\(36 - 16 = 20\)
The difference between consecutive terms is not constant (12 \(\neq\) 20). Therefore, this is not an A.P.

Step 4: Final Answer:
The sequence that is not in an A.P. is \(2^2, 4^2, 6^2, 8^2, ...\).