In the number series 3, 9, 19, 33, 51, the next term is _______. – UGC NET Question

The series can be generated using the expression 2n² + 1 for consecutive integers n. For n = 1 to 5 we obtain 2(1)² + 1 = 3, 2(2)² + 1 = 9, 2(3)² + 1 = 19, 2(4)² + 1 = 33 and 2(5)² + 1 = 51. The next integer is 6, so the next term should be 2(6)² + 1. This equals 2 × 36 + 1 = 72 + 1 = 73, which is consistent with the pattern.

Option A:
Option A gives 73, which exactly matches the formula 2n² + 1 when n = 6. The extended sequence 3, 9, 19, 33, 51, 73 is generated smoothly from one algebraic rule. This confirms that 73 is the correct continuation of the number series.

Option B:
Option B suggests 71, which does not satisfy the expression 2n² + 1 for any integer following 5 in this context. Using 71 would break the simple polynomial structure underlying the series. Therefore, 71 cannot be the correct next term.

Option C:
Option C provides 75, which also fails to match 2n² + 1 for the next integer. It introduces an inconsistent jump that is not supported by the existing values. Hence, 75 is not in harmony with the pattern of the sequence.

Option D:
Option D offers 77, which similarly does not correspond to 2n² + 1 for n = 6 or any immediate extension. Including 77 would abandon the clear quadratic rule. Thus, 77 is not the appropriate choice.