Question:
Which of the following sequence is not an A.P. ?
Which of the following sequence is not an A.P. ?
Show Hint
A sequence of squares of consecutive odd integers (or consecutive natural numbers) never forms an A.P. because the gap between consecutive squares increases quadratically.
This allows you to quickly identify option (D) as the correct choice without doing any calculations!
This allows you to quickly identify option (D) as the correct choice without doing any calculations!
Updated On: Jul 7, 2026
- \(2, \frac{5}{2}, 3, \frac{7}{2}, \dots\)
- \(-1.2, -3.2, -5.2, -7.2, \dots\)
- \(\sqrt{2}, \sqrt{8}, \sqrt{18}, \dots\)
- \(1^2, 3^2, 5^2, 7^2, \dots\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
We are given four sequences of numbers. We need to identify which of these sequences does
not form an Arithmetic Progression (A.P.).
Step 2: Key Formula or Approach:
A sequence forms an A.P. if the difference between any two consecutive terms remains constant throughout the sequence:
\[ d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 \]
We will evaluate the common difference for each option.
Step 3: Detailed Explanation:
1.
Analyze Option (A): \(2, \frac{5}{2}, 3, \frac{7}{2}, \dots\)
- Let's check differences:
\[ a_2 - a_1 = \frac{5}{2} - 2 = 0.5 \]
\[ a_3 - a_2 = 3 - \frac{5}{2} = 0.5 \]
\[ a_4 - a_3 = \frac{7}{2} - 3 = 0.5 \]
Since the difference is constant (\(d = 0.5\)), this is an A.P.
2.
Analyze Option (B): \(-1.2, -3.2, -5.2, -7.2, \dots\)
- Check differences:
\[ a_2 - a_1 = -3.2 - (-1.2) = -2.0 \]
\[ a_3 - a_2 = -5.2 - (-3.2) = -2.0 \]
Since the difference is constant (\(d = -2\)), this is an A.P.
3.
Analyze Option (C): \(\sqrt{2}, \sqrt{8}, \sqrt{18}, \dots\)
- Let's simplify the radical terms:
\[ a_1 = \sqrt{2} \]
\[ a_2 = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
\[ a_3 = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]
- Check differences:
\[ a_2 - a_1 = 2\sqrt{2} - \sqrt{2} = \sqrt{2} \]
\[ a_3 - a_2 = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \]
Since the difference is constant (\(d = \sqrt{2}\)), this is an A.P.
4.
Analyze Option (D): \(1^2, 3^2, 5^2, 7^2, \dots\)
- Let's evaluate the square terms:
\[ 1, 9, 25, 49, \dots \]
- Check differences:
\[ a_2 - a_1 = 9 - 1 = 8 \]
\[ a_3 - a_2 = 25 - 9 = 16 \]
Since \(8 \neq 16\), the differences are not constant. Therefore, this sequence does not form an A.P.
Step 4: Final Answer:
The sequence \(1^2, 3^2, 5^2, 7^2, \dots\) is not an A.P., which corresponds to option (D).
We are given four sequences of numbers. We need to identify which of these sequences does
not form an Arithmetic Progression (A.P.).
Step 2: Key Formula or Approach:
A sequence forms an A.P. if the difference between any two consecutive terms remains constant throughout the sequence:
\[ d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 \]
We will evaluate the common difference for each option.
Step 3: Detailed Explanation:
1.
Analyze Option (A): \(2, \frac{5}{2}, 3, \frac{7}{2}, \dots\)
- Let's check differences:
\[ a_2 - a_1 = \frac{5}{2} - 2 = 0.5 \]
\[ a_3 - a_2 = 3 - \frac{5}{2} = 0.5 \]
\[ a_4 - a_3 = \frac{7}{2} - 3 = 0.5 \]
Since the difference is constant (\(d = 0.5\)), this is an A.P.
2.
Analyze Option (B): \(-1.2, -3.2, -5.2, -7.2, \dots\)
- Check differences:
\[ a_2 - a_1 = -3.2 - (-1.2) = -2.0 \]
\[ a_3 - a_2 = -5.2 - (-3.2) = -2.0 \]
Since the difference is constant (\(d = -2\)), this is an A.P.
3.
Analyze Option (C): \(\sqrt{2}, \sqrt{8}, \sqrt{18}, \dots\)
- Let's simplify the radical terms:
\[ a_1 = \sqrt{2} \]
\[ a_2 = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
\[ a_3 = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]
- Check differences:
\[ a_2 - a_1 = 2\sqrt{2} - \sqrt{2} = \sqrt{2} \]
\[ a_3 - a_2 = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \]
Since the difference is constant (\(d = \sqrt{2}\)), this is an A.P.
4.
Analyze Option (D): \(1^2, 3^2, 5^2, 7^2, \dots\)
- Let's evaluate the square terms:
\[ 1, 9, 25, 49, \dots \]
- Check differences:
\[ a_2 - a_1 = 9 - 1 = 8 \]
\[ a_3 - a_2 = 25 - 9 = 16 \]
Since \(8 \neq 16\), the differences are not constant. Therefore, this sequence does not form an A.P.
Step 4: Final Answer:
The sequence \(1^2, 3^2, 5^2, 7^2, \dots\) is not an A.P., which corresponds to option (D).
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