In the number series 7, 35, 111, 259, 503, the next term is _______. – UGC NET Question

The nth term of this sequence is given by aₙ = 4n³+3 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 4·1³+3 = 7, 4·2³+3 = 35, 4·3³+3 = 111, 4·4³+3 = 259 and 4·5³+3 = 503. For n = 6 the same rule yields 4·6³+3 = 864+3 = 867. Hence 867 is the only number that continues the sequence according to this cubic formula.

Option A:
Option A, 867, comes directly from applying aₙ = 4n³+3 when n = 6. It maintains exactly the same relationship between the index and the term that holds for all earlier values. Because no change in the rule is needed, 867 is fully consistent with the pattern and is correct.

Option B:
Option B, 827, is significantly smaller than 867 and does not equal 4·6³+3. It would require subtracting 40 from the formula’s result with no justification from the series. Therefore 827 is not a valid next term.

Option C:
Option C, 843, is closer but still fails to match the computed value, and it is not the output of 4n³+3 for any integer following n = 5. Selecting it would disrupt the clean cubic structure of the sequence. Thus 843 cannot be the correct answer.

Option D:
Option D, 897, overshoots the value given by the formula and similarly cannot be derived from 4n³+3 for n = 6. Using 897 would destroy the exact fit between the rule and the data, so it is not appropriate.