Question

Which of the following is in an A.P. ?

• A. 1, 7, 9, 16, ...
• B. \(x^2, x^3, x^4, x^5, ...\)
• C. x, 2x, 3x, 4x, ...
• D. \(2^2, 4^2, 6^2, 8^2, ...\)

Hint:

To quickly identify an A.P., simply subtract the first term from the second, and the second from the third. If these two results are not equal, you can immediately rule it out.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
An Arithmetic Progression (A.P.) is a sequence where the difference between any two consecutive terms is constant. This constant value is called the common difference.

Step 2: Key Formula or Approach:
To check if a sequence is an A.P., we calculate the difference between consecutive terms. If \(a_2-a_1 = a_3-a_2 = a_4-a_3 = \dots\), then the sequence is an A.P.

Step 3: Detailed Explanation:
Let's check each option:
(A) 1, 7, 9, 16, ...
\(a_2 - a_1 = 7 - 1 = 6\)
\(a_3 - a_2 = 9 - 7 = 2\)
Since the differences are not constant (6 \(\neq\) 2), this is not an A.P.
(B) \(x^2, x^3, x^4, x^5, ...\)
\(a_2 - a_1 = x^3 - x^2 = x^2(x - 1)\)
\(a_3 - a_2 = x^4 - x^3 = x^3(x - 1)\)
The differences are not constant (unless x=0 or x=1), so this is not an A.P. in general. It is a Geometric Progression.
(C) x, 2x, 3x, 4x, ...
\(a_2 - a_1 = 2x - x = x\)
\(a_3 - a_2 = 3x - 2x = x\)
\(a_4 - a_3 = 4x - 3x = x\)
Since the difference is constant (\(x\)), this sequence is an A.P.
(D) \(2^2, 4^2, 6^2, 8^2, ...\) which is 4, 16, 36, 64, ...
\(a_2 - a_1 = 16 - 4 = 12\)
\(a_3 - a_2 = 36 - 16 = 20\)
Since the differences are not constant (12 \(\neq\) 20), this is not an A.P.

Step 4: Final Answer:
The sequence that is in an A.P. is x, 2x, 3x, 4x, ....