In the mixed number series 3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, ..., what number should come next to correctly continue the pattern?

Introduction:
This question presents a number series that is not a simple arithmetic or geometric progression. Instead, it is constructed from known Pythagorean triples. Recognizing this structure is essential for finding the next term in the sequence.

Given Data / Assumptions:

• The series is: 3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, ...
• We need to identify the pattern and determine the next number.

Concept / Approach:
Group the numbers to see if they form meaningful sets. Notice that they can be grouped as triples:
(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, ...)Each of the first three groups is a well-known Pythagorean triple where a^2 + b^2 = c^2.

Step-by-Step Solution:
Step 1: Identify the triples.First triple: 3, 4, 5 → 3^2 + 4^2 = 9 + 16 = 25 = 5^2.Second triple: 5, 12, 13 → 5^2 + 12^2 = 25 + 144 = 169 = 13^2.Third triple: 7, 24, 25 → 7^2 + 24^2 = 49 + 576 = 625 = 25^2.Step 2: Look at the next group.Numbers so far: 8, 15, ...We recognise another Pythagorean triple: 8, 15, 17.8^2 + 15^2 = 64 + 225 = 289 = 17^2.Step 3: Conclude the next term.The missing number after 8, 15 should be 17.

Verification / Alternative check:
The pattern consistently follows primitive Pythagorean triples: (3, 4, 5), (5, 12, 13), (7, 24, 25), and now (8, 15, 17). There is no break or inconsistency in this structure, so the triple (8, 15, 17) naturally completes the next group.

Why Other Options Are Wrong:
26, 16, 18 and 20 do not complete a standard Pythagorean triple with 8 and 15. Only 17 satisfies 8^2 + 15^2 = 17^2, so only 17 fits the observable pattern of the series.

Common Pitfalls:
Some learners try to look at differences between terms or use arithmetic progressions, which do not work here. Recognizing well-known triples is a major advantage in such pattern questions.

Final Answer:
The next number in the series is 17.