In the number series 2, 6, 15, 40, 145, the next term is _______. – UGC NET Question

In this sequence each term is given by aₙ = n! + n² for n starting from 1. Checking this rule, we get 1!+1² = 2, 2!+2² = 6, 3!+3² = 15, 4!+4² = 40 and 5!+5² = 145, which matches the given terms exactly. For n = 6 the expression yields 6!+6² = 720+36 = 756. Consequently, 756 is the only term that continues the factorial-plus-square pattern.

Option A:
Option A, 720, corresponds to 6! alone and omits the n² part of the expression. It does not follow the rule aₙ = n!+n² that has generated all previous terms. Therefore option A ignores part of the pattern and cannot be accepted as correct.

Option B:
Option B, 744, adds only 24 to 720 instead of the required 36, so it does not equal 6!+6². Selecting 744 would require changing the square component from n² to a smaller value, which is inconsistent with the earlier terms. Hence option B is not a valid continuation.

Option C:
Option C, 756, matches exactly the value of 6!+6² and maintains both components of the rule: the factorial growth and the quadratic correction. It fits the behaviour observed for n = 1 to 5, where each term is one factorial plus a square. Because it preserves this combined structure, 756 is the correct next term in the series.

Option D:
Option D, 780, adds 60 to 720 and cannot be written as n!+n² for n = 6. It would imply using a different function of n for the second part of the expression. This breaks the pattern evident from the given terms, so option D is incorrect.