Consider the number series: 16, 33, 65, 131, 261, ( ? ). Following the same pattern, what is the next term in this sequence?

Introduction / Context:
Number series questions test your ability to recognize patterns and rules in sequences. Here, we are given a series of five numbers and asked to predict the sixth term. To solve this type of problem, you must observe how each term is related to the previous ones, often through addition, subtraction, multiplication, or a combination of these operations.

Given Data / Assumptions:

• Given series: 16, 33, 65, 131, 261, ?
• We assume there is a single consistent pattern for the entire sequence.
• We must find the missing sixth term.

Concept / Approach:
A common approach is to look at the differences between consecutive terms. If the difference itself follows a pattern (for example, doubling, arithmetic progression, or alternating additions and subtractions), we can extend this pattern to find the next term. Sometimes the series might follow a rule like a(n) = 2 * a(n-1) plus or minus a small number.

Step-by-Step Solution:
Step 1: Compute first-level differences between consecutive terms. 16 to 33: 33 - 16 = 17. 33 to 65: 65 - 33 = 32. 65 to 131: 131 - 65 = 66. 131 to 261: 261 - 131 = 130. Step 2: List the differences: 17, 32, 66, 130. Step 3: Notice that each difference is approximately double the previous one, with a small alternating adjustment. Double 17 and subtract 2: 2 * 17 - 2 = 34 - 2 = 32. Double 32 and add 2: 2 * 32 + 2 = 64 + 2 = 66. Double 66 and subtract 2: 2 * 66 - 2 = 132 - 2 = 130. Step 4: The pattern alternates: -2, +2, -2, so the next step should be +2. Double 130 and add 2: 2 * 130 + 2 = 260 + 2 = 262. Step 5: Add this new difference to the last known term: 261 + 262 = 523.

Verification / Alternative Check:
Check consistency: If we write all differences as per the discovered rule, they are 17, 32, 66, 130, 262. Each can be obtained from the previous one using the alternating pattern. Since the logic holds for every step, our prediction of the next term as 523 is reliable.

Why Other Options Are Wrong:
521, 613 and 721 do not fit the carefully derived difference pattern. Using any of these would break the alternating rule on the differences and would not maintain the progression we observed.

Common Pitfalls:
A typical mistake is to look only for simple differences or ratios and give up when a perfect arithmetic or geometric progression is not found. Many exam series use second-level patterns, such as patterns within the differences themselves. Always consider both the terms and their differences before deciding on the rule.

Final Answer:
The next term in the series is 523.