In the number series 16, 33, 65, 131, 261, ( ? ), which value should replace the missing term to maintain the pattern?

Introduction / Context:
This question asks you to insert the missing number in a series where each term appears to be roughly double the previous one, with a small correction. This is a typical number-series puzzle that evaluates your ability to analyze how each term is generated from its predecessor.

Given Data / Assumptions:
The given series is: 16, 33, 65, 131, 261, ( ? ). You must decide which of the options 523, 521, 613 or 721 fits the pattern. We assume that a simple arithmetic relationship of the form “multiply by 2 and then add or subtract 1” is used repeatedly.

Concept / Approach:
When numbers nearly double at each step but do not match exact powers of 2, it is natural to test the rule: next term = 2 * current term ± 1. Here, we will check whether an alternating +1, -1 pattern is followed when doubling each term.

Step-by-Step Solution:
Step 1: Move from 16 to 33. Compute 2 * 16 = 32, then 32 + 1 = 33.Step 2: Move from 33 to 65. Compute 2 * 33 = 66, then 66 - 1 = 65.Step 3: Move from 65 to 131. Compute 2 * 65 = 130, then 130 + 1 = 131.Step 4: Move from 131 to 261. Compute 2 * 131 = 262, then 262 - 1 = 261.Step 5: The pattern is now clear: starting from 16, each term is twice the previous one, alternately adding 1, then subtracting 1. The signs are +1, -1, +1, -1, and so on.Step 6: To get the next term from 261, we double it and then add 1 (because the last operation was -1, so the next must be +1): 2 * 261 = 522, and 522 + 1 = 523.

Verification / Alternative check:
We can reconstruct the entire pattern explicitly: 16 * 2 + 1 = 33, 33 * 2 - 1 = 65, 65 * 2 + 1 = 131, 131 * 2 - 1 = 261, 261 * 2 + 1 = 523. No other option fits this simple and consistent rule. Values like 521 or 613 would break the neat alternation of +1 and -1 after doubling.

Why Other Options Are Wrong:
521 differs from 523 by 2 and cannot be obtained from 261 using a simple double-and-add-or-subtract-1 pattern. Similarly, 613 and 721 are far away from 2 * 261, so they would require arbitrary operations that do not match the earlier steps of the series. Therefore, none of them can be correct.

Common Pitfalls:
Many candidates notice the approximate doubling but ignore the small corrections. Others may try to fit a more complex polynomial pattern instead of testing the straightforward rule “2n ± 1.” Recognizing small consistent adjustments is crucial for solving such problems quickly.

Final Answer:
The missing number that keeps the pattern intact is 523.