Problem 19 · 2019 Math Kangaroo
Hard
Algebra & Patterns
estimate-and-picksubstitution
What is the biggest integer smaller than \(\sqrt{20 + \sqrt{20 + \sqrt{20 + \sqrt{20 + \sqrt{20}}}}}\)?
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Answer: A — 4
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Hint 1 of 2
If the nesting went on forever, the value would be just slightly larger than this finite one.
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Hint 2 of 2
Solve \(x = \sqrt{20 + x}\) for the limiting value, then note the finite version is a little smaller and take its floor.
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Approach: compare the nested radical with its infinite limit
- If the nesting continued forever, the value \(x\) would satisfy \(x = \sqrt{20 + x}\), i.e. \(x^2 - x - 20 = 0\), giving \(x = 5\).
- The actual expression has only finitely many layers, so it is strictly less than 5.
- Numerically it is about \(4.99995\), so the biggest integer smaller than it is 4.
- Answer (A) 4.
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