The sum of the squares of three consecutive integers is 365. What are the integers? - Coaching Toolbox
Curious Minds Ask: The sum of the squares of three consecutive integers is 365. What are the integers?
Curious Minds Ask: The sum of the squares of three consecutive integers is 365. What are the integers?
Why are so many people quietly solving a number puzzle online? Right now, curiosity about unexpected math patterns is skyrocketing—especially around problems like the sum of the squares of three consecutive integers is 365. What are the integers? This question is more than a classroom puzzle; it reflects a growing interest in logic, patterns, and real-world applications of math. As students, educators, and independent learners explore these kinds of problems, they uncover deeper connections to algebra, number theory, and even cryptography.
Understanding how this equation works not only satisfies intellectual curiosity but also sharpens analytical thinking—skills increasingly valued in a data-driven economy. The query taps into a broader trend where math is no longer seen as abstract, but as a practical tool for problem-solving and pattern recognition across technology, finance, and everyday logic.
Understanding the Context
How Does the Sum of Three Consecutive Inte tribute Square to 365?
To solve the mystery: Let the smallest of the three consecutive integers be n. The next two would be n+1 and n+2. Their squares are:
n² + (n+1)² + (n+2)²
Expanding each term gives:
n² + (n² + 2n + 1) + (n² + 4n + 4)
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Combining terms results in:
3n² + 6n + 5
Set this equal to 365:
3n² + 6n + 5 = 365
→ 3n² + 6n – 360 = 0
→ n² + 2n – 120 = 0
Factoring or using the quadratic formula:
n = [–2 ± √(4 + 480)] ÷ 2 = [–2 ± √484] ÷ 2 = [–2 ± 22] ÷ 2
Positive solution: (20) ÷ 2 = 10
So the integers are 10, 11, and 12.
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