In the number series 2, 8, 26, 62, 122, the next term is _______. – UGC NET Question

The terms can be generated by the formula aₙ = n³−n+2 with n starting from 1. For n = 1, 2, 3, 4 and 5 we have 1³−1+2 = 2, 2³−2+2 = 8, 3³−3+2 = 26, 4³−4+2 = 62 and 5³−5+2 = 122, which matches the sequence. For n = 6 we get 6³−6+2 = 216−6+2 = 212. Thus 212 is the unique next term that preserves this cubic-minus-linear pattern.

Option A:
Option A, 202, is 10 less than the value predicted by the formula for n = 6. It is not equal to 6³−6+2 and therefore cannot be justified by the same rule. Choosing 202 would disrupt the precise algebraic structure of the sequence.

Option B:
Option B, 206, also differs from 212 and does not satisfy the relationship aₙ = n³−n+2 for the next index. It would amount to an arbitrary adjustment with no support from the earlier terms. Hence 206 is not the correct continuation.

Option C:
Option C, 220, is larger than the correct cubic value and similarly fails to equal 6³−6+2. Using 220 would break the pattern that neatly connects all the preceding terms to their positions. For this reason, 220 is not a valid choice.

Option D:
Option D, 212, is exactly the result of applying the formula with n = 6. It continues the same rule without any modification and keeps the entire series logically coherent. Therefore this option is the correct next term.