In the number series 4, 20, 46, 82, 128, the next term is _______. – UGC NET Question

This series is described by the formula aₙ = 5n²+n−2 for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 5·1²+1−2 = 4, 5·2²+2−2 = 20, 5·3²+3−2 = 46, 5·4²+4−2 = 82 and 5·5²+5−2 = 128. For n = 6 the same expression gives 5·6²+6−2 = 180+6−2 = 184. Therefore 184 is the correct next term under this quadratic rule.

Option A:
Option A, 174, is 10 less than the value predicted by 5n²+n−2 for n = 6. It does not fit the formula and would require arbitrarily lowering the computed term. Hence 174 is not consistent with the sequence’s structure.

Option B:
Option B, 184, exactly equals 5·6²+6−2 and so preserves the quadratic relationship between n and aₙ. It ensures that the same rule that explains all previous terms also generates the new one. For this reason, 184 is the correct continuation.

Option C:
Option C, 178, also deviates from the calculated value and does not arise from the given expression. It would amount to changing the pattern only at the last step, which is logically unsound. Thus 178 cannot be the right answer.

Option D:
Option D, 192, is larger than the formula’s result and again cannot be written as 5n²+n−2 for n = 6. Selecting 192 would break the precise connection between position and term, so it is not valid.