Find the missing term in the series given below: 1, 7, 25, 61, 121, ?, 337

Concept

A number series can be tested by examining its consecutive differences. For a sequence defined by a cubic expression, those differences follow a quadratic pattern.

For an = n3 − n + 1, the difference an+1 − an equals 3n(n + 1), so the successive additions are 6, 18, 36, 60, 90 and 126. The superscript 3 denotes cubing, while the subscripts identify term positions.

Application

1. The consecutive differences already visible are 7 − 1 = 6, 25 − 7 = 18, 61 − 25 = 36 and 121 − 61 = 60.

2. The next addition is 3 × 5 × 6 = 90, so the missing term is 121 + 90 = 211.

3. The following addition is 3 × 6 × 7 = 126, and 211 + 126 = 337, which reproduces the final term.

Cross-check

Using the direct rule, a6 = 63 − 6 + 1 = 211 and a7 = 73 − 7 + 1 = 337.

Therefore, the missing term is 211.