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

To determine which sequence is not an arithmetic progression (A.P.), we need to check if the sequence has a common difference between its consecutive terms. An A.P. is a sequence of numbers in which the difference of any two successive members is a constant. 

1. Option 1: \(2, \frac{5}{2}, 3, \frac{7}{2}, ...\)
   • The first term \(a_1 = 2\).
   • The second term \(a_2 = \frac{5}{2}\). The difference: \(a_2 - a_1 = \frac{5}{2} - 2 = \frac{1}{2}\).
   • The third term \(a_3 = 3\). The difference: \(a_3 - a_2 = 3 - \frac{5}{2} = \frac{1}{2}\).
   • The third difference \((a_4 = \frac{7}{2} - a_3 = \frac{1}{2})\) is the same as the previous differences, indicating a common difference.
   • This sequence is an A.P.
2. Option 2: \(-1.2, -3.2, -5.2, -7.2, ...\)
   • The first term \(a_1 = -1.2\).
   • The second term \(a_2 = -3.2\). The difference: \(a_2 - a_1 = -3.2 + 1.2 = -2.0\).
   • The third term \(a_3 = -5.2\). The difference: \(a_3 - a_2 = -5.2 + 3.2 = -2.0\).
   • Similar difference for the fourth term:\(-2.0\).
   • This sequence is an A.P.
3. Option 3: \(\sqrt{2}, \sqrt{8}, \sqrt{18}, ...\)
   • The first term \(a_1 = \sqrt{2}\).
   • The second term \(a_2 = \sqrt{8} = 2\sqrt{2}\). The difference: \(a_2 - a_1 = 2\sqrt{2} - \sqrt{2} = \sqrt{2}\).
   • The third term \(a_3 = \sqrt{18} = 3\sqrt{2}\). The difference: \(a_3 - a_2 = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}\).
   • This sequence is an A.P.
4. Option 4: \(1^2, 3^2, 5^2, 7^2, ...\)
   • The first term \(a_1 = 1^2 = 1\).
   • The second term \(a_2 = 3^2 = 9\). The difference: \(a_2 - a_1 = 9 - 1 = 8\).
   • The third term \(a_3 = 5^2 = 25\). The difference: \(a_3 - a_2 = 25 - 9 = 16\).
   • The difference is not constant (\(8 \neq 16\)).
   • This sequence is not an A.P.

Therefore, the sequence that is not an arithmetic progression (A.P.) is \(1^2, 3^2, 5^2, 7^2, ...\).