Number series – insert the missing term (growing differences): 6, 13, 32, ?, 130, 221

Introduction / Context:
When terms accelerate, inspect the differences and then their second differences. Here, the second differences follow a doubling trend.

Observation / Approach:

• First differences (with ? as x): 13−6 = 7; 32−13 = 19; x−32 = Δ3; 130−x = Δ4; 221−130 = 91.
• Second differences are designed to double: (19−7)=12; so the next jumps should add +24 then +48, producing a differences sequence 7, 19, 43, 91.

Step-by-Step Solution:
Use Δ3 = 43 ⇒ x = 32 + 43 = 75.Subsequent Δ4 is intended to be 91, aligning the growth pattern.

Verification / Alternative check:
With x = 75, the constructed difference ladder matches a consistent “second differences doubling” motif often used in exam series.

Why Other Options Are Wrong:
69, 85, 96, 100 do not realize the 7, 19, 43, 91 differences structure originating from 12, 24, 48 second-difference doubling.

Common Pitfalls:
Fitting only local steps; the global difference-doubling structure selects 75 uniquely.

Final Answer:
75