Complete the next number in the given number series: 21, 56, 102, 160, 231, 316, ?

Introduction / Context:
This series increases in a non linear way, which suggests that the differences between terms themselves may follow a pattern. When differences between consecutive terms are not constant, second differences often reveal a simple arithmetic rule. Questions of this type test your understanding of first and second differences in sequences, a common technique in advanced number series problems.

Given Data / Assumptions:

• Series: 21, 56, 102, 160, 231, 316, ?
• Exactly one next term is missing.
• The pattern is expected to be systematic, likely involving changing differences between terms.

Concept / Approach:
First compute the differences between consecutive terms. If those differences do not form a constant sequence, compute the differences of the differences (second differences). A constant or steadily changing second difference sequence usually indicates that the original series is built from a quadratic type pattern. We can then extend the second difference pattern to get the next first difference and hence the missing term.

Step-by-Step Solution:
Step 1: Compute the first differences. 56 - 21 = 35. 102 - 56 = 46. 160 - 102 = 58. 231 - 160 = 71. 316 - 231 = 85. Step 2: Collect these differences: 35, 46, 58, 71, 85. Step 3: Compute the second differences (differences of the first differences). 46 - 35 = 11. 58 - 46 = 12. 71 - 58 = 13. 85 - 71 = 14. Step 4: Second differences follow the simple pattern 11, 12, 13, 14, so the next one should be 15. Step 5: The next first difference will therefore be 85 + 15 = 100. Step 6: Add this new first difference to the last known term: 316 + 100 = 416.

Verification / Alternative check:
Extend the series with the found term: 21, 56, 102, 160, 231, 316, 416. Recompute the first differences: 35, 46, 58, 71, 85, 100. Second differences then become 11, 12, 13, 14, 15, which is a perfectly regular sequence increasing by 1 each time. This confirms that the structure of the series is maintained and validates 416 as the correct missing number.

Why Other Options Are Wrong:
If we try 415, 414, or 413 as the final term, the last first difference and the corresponding second difference no longer follow the neat pattern. For example, if we used 415, the last difference would be 415 - 316 = 99, giving a second difference of 99 - 85 = 14, which repeats rather than increasing to 15. Since the question clearly follows an increasing second difference sequence, these alternative values cannot be correct.

Common Pitfalls:
Many students stop after examining only first differences and conclude that the series is irregular. In such cases, it is crucial to remember the concept of second differences, which often reveals a much simpler hidden pattern. Another mistake is to guess a number based on rough growth estimation instead of verifying it through systematic calculations.

Final Answer:
The next number that correctly completes the series is 416.