In the number series 3, 56, 463, 2004, 6179, the next term is _______. – UGC NET Question

The pattern here is aₙ = 2n⁵ − 3n² + 4 with n beginning at 1. For n = 1, 2, 3, 4 and 5 this expression gives 3, 56, 463, 2004 and 6179, exactly reproducing the given sequence. When n = 6 we get a₆ = 2·6⁵ − 3·6² + 4 = 2·7776 − 108 + 4 = 15552 − 104 = 15448. Thus 15448 is the unique value consistent with this quintic pattern.

Option A:
Option A, 15448, is exactly the outcome of substituting n = 6 into 2n⁵ − 3n² + 4. It maintains the same dominance of the fifth-power term with a negative quadratic adjustment and a constant shift. Because this structure accounts for all earlier terms and extends naturally to 15448, this option correctly continues the series.

Option B:
Option B, 15424, is 24 less than the rule’s value and cannot be generated by the same polynomial at n = 6. Accepting 15424 would require reducing the correct result arbitrarily at the final term. This breaks the algebraic coherence of the sequence, so option B is incorrect.

Option C:
Option C, 15436, is 12 less than 15448 and again does not equal 2·6⁵ − 3·6² + 4. It represents an approximate but inaccurate candidate that does not come from the established rule. Therefore option C is not a valid next term.

Option D:
Option D, 15460, is 12 greater than the computed value and would demand increasing the polynomial output only at this step. Since earlier terms exactly match the expression, this modification is unjustified. Hence option D does not correctly extend the pattern.