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

To determine which of the given sequences is not an arithmetic progression (A.P.), we must first understand that an arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference.

1. \(2, \frac{5}{2}, 3, \frac{7}{2}, \dots\)
    

Let's find the common difference:

• Difference between the second term and the first term: \(\frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2}\)
• Difference between the third term and the second term: \(3 - \frac{5}{2} = \frac{6}{2} - \frac{5}{2} = \frac{1}{2}\)

1. \(-1.2, -3.2, -5.2, -7.2, \dots\)
    

Let's find the common difference:

• Difference between the second term and the first term: \(-3.2 - (-1.2) = -2\)
• Difference between the third term and the second term: \(-5.2 - (-3.2) = -2\)

1. \(\sqrt{2}, \sqrt{8}, \sqrt{18}, \dots\)
    

Let's check for a common difference:

• Difference between the second term and the first term: \(\sqrt{8} - \sqrt{2} \neq \text{constant}\)
• Difference between the third term and the second term: \(\sqrt{18} - \sqrt{8} \neq \text{constant}\)

1. \(1^2, 3^2, 5^2, 7^2, \dots\)
    

These are the squares of the sequence \(1, 3, 5, 7, \dots\) which indeed forms an A.P. with common difference 2. However, the sequence of their squares does not form an A.P.:

• Calculate: \(1^2 = 1, 3^2 = 9, 5^2 = 25, 7^2 = 49\)
• Difference: \(9 - 1 = 8,\ 25 - 9 = 16,\ 49 - 25 = 24\)

Upon comparison, the sequence that is not an A.P. is \(1^2, 3^2, 5^2, 7^2, \dots\).