In this number series aptitude question, determine the next term in the sequence: 1, 3, 6, 11, 20, 37, ?

Introduction / Context:
This number series problem checks your ability to recognize patterns in second-level differences. Instead of a simple arithmetic or geometric progression, the series 1, 3, 6, 11, 20, 37, ? is built using a rule that applies to the differences between consecutive terms, and those differences themselves follow another pattern.

Given Data / Assumptions:

• Series: 1, 3, 6, 11, 20, 37, ?
• Exactly one correct next term needs to be found.
• The pattern is consistent throughout the series.
• The same logic that produced earlier terms is used to produce the next term.

Concept / Approach:
When the pattern in the terms is not immediately obvious, a standard method is to examine the differences between consecutive terms. Sometimes these differences form a new recognizable series, such as an arithmetic progression or a pattern involving multiplication or doubling. If needed, we then check the differences of those differences, known as second-level differences.

Step-by-Step Solution:
Step 1: Compute first-level differences.3 - 1 = 26 - 3 = 311 - 6 = 520 - 11 = 937 - 20 = 17So the differences are: 2, 3, 5, 9, 17.Step 2: Look for a pattern in these differences.Observe that each difference from the second one onward is obtained using the rule: next difference = 2 * previous difference - 1.Check: 2 * 2 - 1 = 3, 2 * 3 - 1 = 5, 2 * 5 - 1 = 9, 2 * 9 - 1 = 17.Step 3: Use the same rule to get the next difference.Next difference = 2 * 17 - 1 = 34 - 1 = 33.Step 4: Add this next difference to the last term.Required next term = 37 + 33 = 70.

Verification / Alternative check:
Rebuilding the series with the rule confirms the answer. Starting from 1 and repeatedly adding differences 2, 3, 5, 9, 17, 33 we obtain 1, 3, 6, 11, 20, 37, 70. Every step respects the formula next difference = 2 * previous difference - 1, so the pattern is consistent.

Why Other Options Are Wrong:
49: Would correspond to adding only 12 to 37, which does not fit the rapid growth pattern of the differences.
56: Requires adding 19, but 19 cannot be obtained from the rule 2 * previous difference - 1 applied to 17.
64: Would require adding 27, which does not match the consistent recurrence for differences.

Common Pitfalls:
Students often try to force a simple arithmetic progression on the original terms, missing that the real structure lies in the differences. Another mistake is stopping after checking only one or two differences and guessing. For non-trivial series like this, always consider patterns in the first and sometimes second-level differences.

Final Answer:
The next number in the sequence is 70.