Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
2, 6, 20, 42, 110, 156, 272, .....
Assertion A: In the above number series the number is subtracted from the square of the same prime number.
Reason R: The difference between the two consecutive numbers of the series are multiple of even numbers only.
In the light of the above statements, choose the most appropriate answer from the options given below.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A is correct but R is false
Note: The given answer is as per the official NTA Answer Key. However, the question has two possible answers, i.e. both Option (2) and Option (3), as explained below.
Assertion A — True. Each term is a prime number subtracted from its own square, taking the primes 2, 3, 5, 7, 11, 13, 17 in order:
\(2^2-2=2\), \(3^2-3=6\), \(5^2-5=20\), \(7^2-7=42\)
\(11^2-11=110\), \(13^2-13=156\), \(17^2-17=272\)
The differences between consecutive terms are 4, 14, 22, 68, 46 and 116. Whether Reason R is true depends on how the word “only” is read.
Reading R strictly (Option 3):
“Multiples of even numbers only” is taken to mean that the differences are multiples of even numbers and of no odd number.
That is false: 14 is a multiple of 7, 22 of 11, 46 of 23 and 116 of 29, all odd numbers. So R is false, and A is correct but R is false.
Reading R plainly (Option 2):
R is taken to say that every difference is an even number, a multiple of 2. All six differences are even, so R is true.
R does not explain A: A gives the rule that makes the terms (\(p^2-p\)), while R only describes the gaps between them. So both are correct, but R is not the correct explanation of A.
Tip for students: When a statement can be read in two ways and each reading gives a different option, mark the one you can defend best, and if NTA’s key gives the other, challenge it when NTA opens the window for challenges.