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
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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