Identify the next number in the following series: 2, 8, 26, 62, 122, 212, ?

Concept: When the first differences between consecutive terms show no obvious pattern, examine the second-order differences (the differences between the first differences). If these second differences themselves grow by a constant amount each step, that constant is the third difference — use it to project the next second difference, then the next first difference, and finally the next term.

1. Find the first differences between consecutive terms: 8-2=6, 26-8=18, 62-26=36, 122-62=60, 212-122=90.

2. Find the second differences (differences of the first differences): 18-6=12, 36-18=18, 60-36=24, 90-60=30.

3. Find the third differences (differences of the second differences): 18-12=6, 24-18=6, 30-24=6 - constant at 6.

4. Since the third difference is constant at 6, project the next second difference: 30+6=36.

5. Project the next first difference using that second difference: 90+36=126.

6. Add this to the last given term: 212+126=338.

Cross-check: the third difference has already held constant at 6 across every consecutive triple of the given terms (12 to 18 to 24 to 30), so extending that same steady growth one more step is consistent with the established trend, confirming 338 as the next term in the series.