In the number series 2, 22, 78, 188, 370, the next term is _______. – UGC NET Question

This sequence is defined by aₙ = 3n³−n for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 3·1³−1 = 2, 3·2³−2 = 22, 3·3³−3 = 78, 3·4³−4 = 188 and 3·5³−5 = 370. For n = 6 the formula yields 3·6³−6 = 648−6 = 642. Thus 642 is the next term that maintains this cubic-minus-linear relationship.

Option A:
Option A, 622, is 20 less than the formula’s outcome and does not equal 3·6³−6. It would require arbitrarily reducing the term, which is not supported by the pattern of the sequence. Therefore 622 cannot be the correct continuation.

Option B:
Option B, 642, matches exactly the result of applying aₙ = 3n³−n for n = 6. It keeps the dependence on the index unchanged and fits neatly after 370. Because the same rule works for all known terms and the next one, 642 is the correct answer.

Option C:
Option C, 630, also deviates from the computed value and is not of the form 3n³−n for the next integer n. Adopting 630 would make the formula inconsistent at the final step. Hence 630 is not valid.

Option D:
Option D, 657, overshoots the correct value and again is not produced by the expression for n = 6. Using 657 would spoil the perfect algebraic fit of the sequence, so it is not correct.