GATE 2020 BM – Question 28
Starting at a finite positive value, the iteration xₙ₊₁=(xₙ+1/xₙ)/2 converges to what value?
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Show answer and explanation
Correct answer: 1 to 1
Explanation
Any positive fixed point satisfies x=(x+1/x)/2, so x²=1 and positivity selects one. After one iteration, the arithmetic–geometric mean inequality gives x≥1. For x≥1, the next iterate lies between one and x, so subsequent iterates decrease to the positive fixed point. The limit is one.