In the number series 6, 17, 34, 57, 86, the next term is _______. – UGC NET Question

The series is described by the quadratic formula aₙ = 3n²+2n+1 for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 6, 17, 34, 57 and 86, which match the given terms exactly. For n = 6 we obtain 3·6²+2·6+1 = 108+12+1 = 121. Hence the next term must be 121 to preserve the same quadratic relationship.

Option A:
Option A, 111, does not equal 3·6²+2·6+1 and therefore fails to extend the polynomial rule consistently. It would require lowering the value obtained from the formula by 10, which has no justification in the structure of the sequence. Thus 111 cannot be the correct continuation.

Option B:
Option B, 121, is exactly the value produced by the formula aₙ = 3n²+2n+1 when n = 6. It maintains the same dependence on n that explains all earlier terms. Because the pattern holds perfectly when this value is used, 121 is the correct next term.

Option C:
Option C, 115, lies between 111 and 121 but does not arise from substituting n = 6 into the quadratic expression. Selecting 115 would break the neat quadratic pattern and make the rule inconsistent. Therefore it is not the right answer.

Option D:
Option D, 129, is larger than the value predicted by the formula and again cannot be written as 3n²+2n+1 for n = 6. Using 129 would lose the tight algebraic link between n and aₙ, so it is not a valid continuation.