In the number series 6, 44, 144, 336, 650, the next term is _______. – UGC NET Question

This series is generated by the formula aₙ = 5n³+n² for n starting from 1. For n = 1, 2, 3, 4 and 5 we have 5·1³+1 = 6, 5·2³+4 = 44, 5·3³+9 = 144, 5·4³+16 = 336 and 5·5³+25 = 650. For n = 6 the same expression yields 5·6³+36 = 1080+36 = 1116. Therefore 1116 is the correct next term that preserves this cubic-plus-square relationship.

Option A:
Option A, 1076, is 40 less than the computed value and does not satisfy aₙ = 5n³+n² for n = 6. It weakens the consistency between the algebraic rule and the numeric sequence. Hence 1076 is not the correct continuation.

Option B:
Option B, 1092, is somewhat closer but still different from 1116 and cannot be obtained from the formula without changing coefficients. This would make the pattern inconsistent at the final term. Thus 1092 is not valid.

Option C:
Option C, 1146, is larger than the correct value and again fails to equal 5n³+n² for n = 6. Choosing 1146 would distort the functional relationship that holds throughout the sequence. Therefore it is not the right answer.

Option D:
Option D, 1116, exactly matches the result given by the rule when n = 6. It keeps the cubic and quadratic contributions aligned in the same way as for earlier terms and thus is the correct next term.