Question : Find the value of $\sqrt{30+\sqrt{30+\sqrt{30+...}}}$ :
Option 1: 5
Option 2: $3\sqrt{10}$
Option 3: 6
Option 4: 7
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Correct Answer: 6
Solution : Let $x = \sqrt{30+\sqrt{30+\sqrt{30+...}}}$ So, ${x = }\sqrt{30+x}$ By squaring both sides, we get, $⇒x^{2}=(\sqrt{30+x})^{2}$ $⇒x^{2}=30+x$ $⇒x^{2}–x–30=0$ $⇒x^{2}– 6x+5x–30=0$ $⇒x(x–6)+5(x–6) = 0$ $⇒(x–6)(x+5)=0$ $⇒x=6,–5$ A square root cannot be negative. Thus, $x$ cannot be –5. So, $x = 6$ Hence, the correct answer is 6.
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