Double then ±1 alternating rule Find the next term: 16, 33, 65, 131, 261, ( ... ).

Introduction / Context:
Some sequences double the current value and then alternately add or subtract 1. Recognizing the alternation (+1, −1, +1, −1, ...) is key to predicting the next figure after a given term.

Given Data / Assumptions:

• Sequence: 16, 33, 65, 131, 261, ...
• Rule suspected: next = (current * 2) ± 1, with signs alternating.

Concept / Approach:
Verify the rule across steps, then apply it once more. The pattern appears to be: ×2 + 1, ×2 − 1, ×2 + 1, ×2 − 1, ...

Step-by-Step Solution:
16 → 33: 16*2 + 1 = 33.33 → 65: 33*2 − 1 = 65.65 → 131: 65*2 + 1 = 131.131 → 261: 131*2 − 1 = 261.261 → next: 261*2 + 1 = 523.

Verification / Alternative check:
Checking sign alternation confirms consistency across all transitions. Therefore, 523 must be the continuation.

Why Other Options Are Wrong:
521 would require −1 (not the correct sign at this step); 613 and 721 do not follow ×2 ± 1; 511 corresponds to a different pattern entirely.

Common Pitfalls:
Missing the ± alternation and applying +1 (or −1) repeatedly, which quickly derails the sequence.

Final Answer:
523