Question:
Three consecutive positive integers are raised to the first, second, and third powers respectively and added. The sum is a perfect square whose square root equals the total of the three original integers. Which range best describes the minimum integer \(m\) of these three?
Three consecutive positive integers are raised to the first, second, and third powers respectively and added. The sum is a perfect square whose square root equals the total of the three original integers. Which range best describes the minimum integer \(m\) of these three?
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Always test small ranges after setting up the algebraic condition to quickly find the valid range.
Updated On: Jul 30, 2025
- \(1 \le m \le 3\)
- \(4 \le m \le 6\)
- \(7 \le m \le 9\)
- \(10 \le m \le 12\)
- \(13 \le m \le 15\)
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The Correct Option is C
Solution and Explanation
Let integers be \(m, m+1, m+2\).
Sum: \(m^1 + (m+1)^2 + (m+2)^3\).
Condition: \(\sqrt{\text{Sum}} = 3m + 3\).
Squaring and solving shows smallest \(m\) satisfying is in range \(7 \le m \le 9\).
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