Question : The value of $\sqrt{72+\sqrt{72+\sqrt{72+...}}}$ is:
Option 1: 9
Option 2: 8
Option 3: 18
Option 4: 12
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Correct Answer: 9
Solution :
Let $ x=\sqrt{72+\sqrt{72+\sqrt{72+...}}}$
$x=\sqrt{72+x}$.
By squaring both sides, we get,
$⇒ x^{2}=(\sqrt{72+x})^{2}$
$ ⇒x^{2}=72+x$
$ ⇒x^{2}-x-72=0$
$ ⇒x^{2}-9x+8x-72=0$
$⇒x(x-9)+8(x-9)=0$
$⇒(x-9)(x+8)=0$
$\therefore x=9,-8$
Since $x$ can not be equal to –8, the value of $x$ will be 9.
Hence, the correct answer is 9.
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