What is the next number in the sequence: 21, 77, 165, 285, ......?

Introduction / Context:
This is a number series question where the differences between successive terms form their own arithmetic progression. Such second order patterns are very common and rely on recognising that the differences themselves follow a simple rule even if the original numbers do not.

Given Data / Assumptions:

• Series: 21, 77, 165, 285, ?
• Exactly one next term must be found.
• The main numbers increase in a non linear way, suggesting a pattern in their differences.

Concept / Approach:
We compute first differences between consecutive terms and then examine whether these differences follow a simple arithmetic progression. If they do, we can extend that progression and then reconstruct the missing term in the original series. This is essentially a second level arithmetic sequence.

Step-by-Step Solution:
Step 1: Compute the first differences. 77 - 21 = 56. 165 - 77 = 88. 285 - 165 = 120. Step 2: Collect these differences: 56, 88, 120. Step 3: Compute the differences of these differences (second differences). 88 - 56 = 32. 120 - 88 = 32. Step 4: The second differences are constant and equal to 32, so the first differences themselves form an arithmetic sequence with common difference 32. Step 5: Extend the first difference sequence: next difference = 120 + 32 = 152. Step 6: Add this to the last known term: 285 + 152 = 437.

Verification / Alternative check:
Write the full pattern of differences: 21, 77 (+56), 165 (+88), 285 (+120), 437 (+152). The first differences 56, 88, 120, 152 form an arithmetic progression with common difference 32. This confirms that 437 is the unique continuation of the series consistent with a constant second difference, which is a standard construction in series questions.

Why Other Options Are Wrong:
If we tried 404, 415, or 426 as the next term and recomputed the differences, the jump from 285 would not be 152, and the resulting first differences would fail to follow an arithmetic progression with common difference 32. Thus these values break the clean pattern of constant second differences and must be rejected.

Common Pitfalls:
Some students stop after observing that first differences are not constant and assume there is no simple pattern. In such cases, it is important to remember the idea of second order differences. Another pitfall is attempting to guess the next term roughly from the size of previous jumps without doing systematic calculations, which can lead to wrong answers when the pattern is precise.

Final Answer:
The next number in the series, preserving the constant second difference of 32, is 437.