In the mixed number pattern 3, 4, 5, 4, 9, 10, 6, 16, 15, 9, 25, 20, 13, 36, ?, identify the missing number that completes the series.

Introduction / Context:
This series is a more complex pattern in which the numbers are grouped in triplets, and different rules may apply to each position within the triplet. Recognizing such structured repetition is a key skill for solving higher level number series problems in reasoning tests.

Given Data / Assumptions:
The series is:
3, 4, 5, 4, 9, 10, 6, 16, 15, 9, 25, 20, 13, 36, ?
We can group the numbers into sets of three:
(3, 4, 5), (4, 9, 10), (6, 16, 15), (9, 25, 20), (13, 36, ?).
We assume that each position within a triplet follows its own consistent pattern across the sequence.

Concept / Approach:
First, examine the second term in each triplet: 4, 9, 16, 25, 36. These are clearly perfect squares of consecutive integers 2, 3, 4, 5, and 6. Next, examine the first terms and the third terms to see whether they follow simpler numeric progressions. Typically, in such questions, one subsequence may be based on squares, another on simple addition, and another on multiples of a constant like 5.

Step-by-Step Solution:
Step 1: Analyze the second terms of each triplet.They are 4, 9, 16, 25, 36 which equal 2^2, 3^2, 4^2, 5^2, 6^2. So second terms are consecutive squares.Step 2: Analyze the first terms of each triplet.They are 3, 4, 6, 9, 13. The differences are 1, 2, 3, and 4. Thus the increments themselves increase by 1.Step 3: Analyze the third terms of each triplet.They are 5, 10, 15, 20, ?. Each is a multiple of 5: 1*5, 2*5, 3*5, 4*5.Step 4: The next term in this subsequence should therefore be 5*5 = 25.Step 5: Hence, the missing term in the full series is 25.

Verification / Alternative check:
Rewriting the series with the observed patterns: the first terms increase with differences 1, 2, 3, and 4 (3, 4, 6, 9, 13), the second terms are squares 2^2 through 6^2, and the third terms are multiples of 5 from 5 to 25. The last triplet is then (13, 36, 25), which fits neatly into the established structure. No contradictions arise, confirming the solution.

Why Other Options Are Wrong:
Values like 17, 28, 31, or 22 are not multiples of 5 and therefore fail to continue the third term subsequence 5, 10, 15, 20. They do not fit the clean pattern where each third term is 5 times a consecutive integer. Only 25 satisfies this requirement.

Common Pitfalls:
Many learners try to treat the entire string of numbers as a single simple series and overlook the grouped structure. This leads to confusion because no straightforward pattern is visible when all numbers are considered together. Always look for repeating blocks or triplets in longer, more complex number sequences.

Final Answer:
The missing number that completes the given series is 25.