Consider the sequence 3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, ... which is built from Pythagorean triples. Which number should come next in place of the dots?

Introduction / Context:
This sequence question hides a pattern based on Pythagorean triples, which are sets of three positive integers (a, b, c) that satisfy a^2 + b^2 = c^2. Recognizing that the numbers are grouped into such triples allows us to find the next term quickly. The problem tests pattern recognition and knowledge of common Pythagorean triples.

Given Data / Assumptions:

- The sequence given is 3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, ... - We suspect the numbers may be grouped into meaningful sets of three. - We must determine the next single number after 8 and 15.

Concept / Approach:
Observe the sequence in groups of three terms. The first group is 3, 4, 5, which is the smallest Pythagorean triple. The second group is 5, 12, 13, another well known triple. The third group is 7, 24, 25, yet another triple. The pattern seems to be that each group of three numbers forms a Pythagorean triple, with the first two being the legs and the third being the hypotenuse. The next group starts with 8 and 15, suggesting we should look for the hypotenuse that completes the triple (8, 15, h).

Step-by-Step Solution:
Step 1: Group the terms as (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, ?). Step 2: Check each triple: 3^2 + 4^2 = 9 + 16 = 25 = 5^2, so (3, 4, 5) is a Pythagorean triple. Step 3: For (5, 12, 13), compute 5^2 + 12^2 = 25 + 144 = 169 = 13^2. Step 4: For (7, 24, 25), compute 7^2 + 24^2 = 49 + 576 = 625 = 25^2. Step 5: The pattern is clear: each group of three consecutive terms forms a Pythagorean triple. Step 6: Now consider (8, 15, ?). Compute 8^2 + 15^2 = 64 + 225 = 289. Step 7: Recognize that 289 is 17^2, since 17 * 17 = 289. Step 8: Therefore, the triple is (8, 15, 17), and the missing next term is 17.

Verification / Alternative check:
We can cross check by listing common Pythagorean triples from memory: (3, 4, 5), (5, 12, 13), (7, 24, 25) and (8, 15, 17). The sequence matches the first three triples exactly and starts the fourth triple with 8 and 15, confirming that 17 must follow. None of the other options fit neatly as a hypotenuse of these legs.

Why Other Options Are Wrong:
Option A (26) would not satisfy 8^2 + 15^2 = 26^2, because 64 + 225 = 289 while 26^2 = 676. Options B (16) and D (18) also do not yield the correct relation when squared and summed. Only 17 gives the correct Pythagorean relationship.

Common Pitfalls:
Some learners may treat the sequence as a simple numeric pattern without grouping the terms into triples, leading to incorrect differences or ratios. Others may not recall common Pythagorean triples and therefore miss the structure. Recognizing triples is a useful skill in many aptitude and geometry problems.

Final Answer:
The next number in the sequence is 17, which corresponds to option C.