Rule-based growth: 4, 25, 130, ?, 3280 Identify the missing term by detecting a consistent “multiply-and-add” pattern.

Introduction / Context:Some series advance by applying the same affine transformation (multiply by a constant, then add a constant). Recognizing this quickly turns the puzzle into a straightforward computation.

Given Data / Assumptions:

• Series: 4, 25, 130, ?, 3280
• A uniform step of the form next = a*current + b is suspected.

Concept / Approach:Try small integer multipliers (e.g., 3, 4, 5) and small additive constants. Validate on the first two jumps; if it matches exactly, extrapolate to the missing middle term and confirm by hitting the final term given.

Step-by-Step Solution:Test *5 + 5: 4*5 + 5 = 25 (fits); 25*5 + 5 = 130 (fits).Apply to find the missing term: 130*5 + 5 = 655.Confirm last step: 655*5 + 5 = 3280 (fits).

Verification / Alternative check:Any alternative affine rule must match three transitions perfectly; *5 + 5 uniquely satisfies all.

Why Other Options Are Wrong:645, 678, and 685 fail to produce 3280 under the same rule on the next step.

Common Pitfalls:Switching rules mid-sequence or trying to force non-integers when a simple small-integer pattern works cleanly.

Final Answer:655