Number series: 2, 15, 41, 80, 132, ? Find the next term by studying second-level patterns (increments of increments).

Introduction / Context:When first differences are not constant but still show a simple structure, the second-level pattern (how the differences change) often reveals the rule.

Given Data / Assumptions:

• Series: 2, 15, 41, 80, 132, ?
• We examine differences between consecutive terms.

Concept / Approach:Compute consecutive differences and look for a linear pattern (e.g., +k, +2k, +3k, …). Then extend that pattern to project the next main term.

Step-by-Step Solution:Differences: 15−2 = 13; 41−15 = 26; 80−41 = 39; 132−80 = 52.These are 13, 26, 39, 52 = 1*13, 2*13, 3*13, 4*13.Next difference = 5*13 = 65.Next term = 132 + 65 = 197.

Verification / Alternative check:The sequence can be reconstructed from 2 by repeatedly adding 13n for n = 1,2,3,4,5…, which unambiguously yields 197 next.

Why Other Options Are Wrong:150, 178, 180 are not 65 more than 132; adopting them would break the clear multiples-of-13 pattern.

Common Pitfalls:Assuming a geometric ratio or adding a fixed number, missing the structured increment growth (second-level linearity).

Final Answer:197