In the number series 122, 197, 290, which of the following numbers should come next to continue the pattern?

Introduction / Context:
This is a numerical series question in which the given numbers are 122, 197, and 290, followed by a missing term. Such series often rely on differences between consecutive numbers, sometimes with those differences themselves following a pattern. Understanding how each term is built from the previous term allows us to predict the next number in the series.

Given Data / Assumptions:

• Given series: 122, 197, 290, ?
• We must find the next number from options 399, 400, 401, 402, 415.
• No external operations are implied; it is purely a pattern based progression.

Concept / Approach:
The standard strategy for number series is to compute the differences between consecutive terms and see whether these differences follow a simple pattern, such as an arithmetic progression or another recognizable sequence. Once a clear pattern is observed in the differences, that pattern is applied to extend the series and determine the missing term. Occasionally, the differences themselves may increase by a constant amount, creating a second level arithmetic progression.

Step-by-Step Solution:
Step 1: Compute the difference between the second and first terms. 197 minus 122 equals 75. Step 2: Compute the difference between the third and second terms. 290 minus 197 equals 93. Step 3: We now have first level differences: 75 and 93. Step 4: Calculate the difference between these two differences. 93 minus 75 equals 18. Step 5: This suggests that the differences are increasing by 18 each time. So the next difference should be 93 plus 18, which equals 111. Step 6: Add this next difference to the last term of the series. 290 plus 111 equals 401. Step 7: Therefore, the next number in the series should be 401.

Verification / Alternative check:
We can interpret 75 and 93 as multiples of 3: 75 is 3 times 25, and 93 is 3 times 31. The jump from 25 to 31 is plus 6. If we increase 31 by 6 again, we get 37. Multiplying by 3 gives 111. This provides a second way to see that 111 is the correct next difference. Adding 111 to 290 yields 401, which matches the answer we obtained earlier. No other option maintains both the increasing difference of 18 and the coincidence with the pattern 3 times an increasing odd number sequence.

Why Other Options Are Wrong:
399 would require a difference of 109 from 290, which does not match the +18 growth from 75 to 93.
400 would involve a difference of 110, also inconsistent with the established second level difference.
402 would require a difference of 112, which is again off by 1 from the expected 111.
415 gives a difference of 125, far away from the predicted pattern. None of these alternatives fits the structured growth of the differences.

Common Pitfalls:
A common mistake is to look only at the absolute sizes of the terms and try arbitrary operations instead of systematically examining first and second level differences. Another pitfall is to assume a constant difference and stop when it does not match, rather than checking whether the differences themselves form a second level arithmetic progression. Careful step by step analysis of differences prevents these errors.

Final Answer:
The next number in the series is 401.