In the number series 3, 6, 11, 18, 27, the next term is _______. – UGC NET Question

Each term can be written as n² + 2 for successive natural numbers n. When n = 1, 2, 3, 4 and 5, we get 1² + 2 = 3, 2² + 2 = 6, 3² + 2 = 11, 4² + 2 = 18 and 5² + 2 = 27. The next integer is 6, so the next term should be 6² + 2. This equals 36 + 2 = 38, which maintains the exact functional rule of the sequence.

Option A:
Option A gives 38, which matches 6² + 2 and extends the pattern generated by the expression n² + 2. The sequence becomes 3, 6, 11, 18, 27, 38, corresponding to n = 1 to 6. This clear algebraic rule supports 38 as the correct next term.

Option B:
Option B offers 36, which is a perfect square but does not fit the formula n² + 2 because 36 would correspond to 6², missing the constant addition of 2. Using 36 would break the specific relationship seen in all earlier terms. Therefore, 36 is not consistent with the governing pattern.

Option C:
Option C gives 40, which is larger than 6² + 2 by 2 and does not correspond to any n² + 2 within the sequence. It appears arbitrary and fails to match the functional rule that defines the series. Hence, 40 cannot be accepted as the correct answer.

Option D:
Option D suggests 42, which also does not align with the expression n² + 2 for the next integer. It introduces a new, unsupported relationship between consecutive terms. Thus, 42 breaks the pattern and is not the correct continuation.