Choose the correct alternative that will complete the given number series. 2,…

Concept

In a cube-based number series, each term is generated from the position number n by a fixed rule of the form n3 plus or minus a constant. The position counts 1, 2, 3, ... for the successive terms. Identify the rule by comparing each given term with the perfect cubes 1, 8, 27, 64, 125, 216.

Application

Match each given term to the cube of its position:

1. 1 + 1 = 1 + 1 = 2 (1st term)

2. 2 + 1 = 8 + 1 = 9 (2nd term)

3. 3 + 1 = 27 + 1 = 28 (3rd term)

4. 4 + 1 = 64 + 1 = 65 (4th term)

5. The rule is therefore n3 + 1. The missing 5th term is 5 + 1 = 125 + 1 = 126.

Cross-check

Apply the same rule to the 6th term: 6 + 1 = 216 + 1 = 217, which matches the last given term. The rule holds across the whole series, confirming 126.

A second check via differences: the gaps 7, 19, 37 grow by 12 then 18 (a constant second-difference increase of 6). The next gap is 37 + 24 = 61, giving 65 + 61 = 126; the following gap 61 + 30 = 91 gives 126 + 91 = 217. Both methods agree.