Question : Find the value of the given expression. $\sqrt{20-\sqrt{20-\sqrt{20-\sqrt{20-\cdots \infty}}}}$
Option 1: 4
Option 2: 6
Option 3: 5
Option 4: 2
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Correct Answer: 4
Solution : Given expression, $\sqrt{20-\sqrt{20-\sqrt{20-\sqrt{20-\cdots \infty}}}}$ Let us equate this with $y$ ⇒ $y=\sqrt{20-\sqrt{20-\sqrt{20-\sqrt{20-\cdots \infty}}}}$ ⇒ $y=\sqrt{20-y}$ Squaring both sides, ⇒ $y^2=20-y$ ⇒ $y^2+y-20=0$ ⇒ $y^2+5y-4y-20=0$ ⇒ $y(y+5)-4(y+5)=0$ ⇒ $(y+5)(y-4)=0$ ⇒ $y=-5$ and $y=4$ Hence, the correct answer is 4.
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