Find the next number in the series: 404, 415, 402, 413, 400, ?

Introduction / Context:
This problem checks understanding of alternating operations in a number series. Instead of using a single subtraction or addition step throughout, many reasoning questions use a repeated pattern such as adding a number, then subtracting a different number, and so on. Here we must find the missing term that continues the alternating pattern in the given sequence of numbers starting from 404.

Given Data / Assumptions:
- Given series: 404, 415, 402, 413, 400, ?- Exactly one term is missing at the end of the series.- The pattern is expected to be consistent between consecutive terms, possibly involving alternation of two fixed operations.

Concept / Approach:
When a series moves up, then down, then up again, a very natural idea is that the operations alternate between addition and subtraction. To reveal the pattern, we compute the differences between consecutive terms and see if they repeat in a regular fashion. Once we identify the two-step cycle, we can apply it to the last known term to get the missing number.

Step-by-Step Solution:
- Difference from 404 to 415: 415 - 404 = 11 (an increase of 11).- Difference from 415 to 402: 415 - 402 = 13 (a decrease of 13).- Difference from 402 to 413: 413 - 402 = 11 (again an increase of 11).- Difference from 413 to 400: 413 - 400 = 13 (a decrease of 13).- We clearly see an alternating pattern: +11, -13, +11, -13, and so on.- To find the next term after 400, we continue the cycle. After a -13 step, the next step is +11.- Therefore, the missing term is 400 + 11 = 411.

Verification / Alternative check:
- Rewriting the complete series with the discovered pattern: 404 (+11) = 415, 415 (-13) = 402, 402 (+11) = 413, 413 (-13) = 400, 400 (+11) = 411.- The pattern of alternating increases and decreases is perfectly preserved when 411 is placed as the sixth term.- No other option allows such a neat and consistent alternation.

Why Other Options Are Wrong:
- 421, 417, 414 and 409 do not give a constant repeating pair of differences of +11 and -13 when substituted at the missing position.- For example, using 409 would give a difference of only +9 from 400, breaking the established pattern.

Common Pitfalls:
- Some test takers look only at approximate differences without checking the exact values and may misread 11 and 13 as the same or similar.- Another pitfall is to try fitting a single constant difference, which does not work because the series clearly moves both upward and downward.- Ignoring the alternation and focusing on more complicated formulas is unnecessary and wastes time in the exam.

Final Answer:
The series alternates between adding 11 and subtracting 13, so the missing number after 400 must be 400 + 11 = 411.