Number theory · Irrationality · Nested radicals · Perfect squares
Prove that the number
\[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \]
is irrational for every natural number \(n \geq 2\).
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Serbian Regional Competition (Okruzno takmicenje) 2002, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source