Question

The value of $\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8}}}}}$

• A. $4$
• B. $6$
• C. $8$
• D. $10$

Answer 1

The correct option is A $4$

$\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8}}}}}$

We can write it as $\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8+....\infty }}}}}$

Let $x=\sqrt{8+2\sqrt{8+2\sqrt{8+\mathrm{2...}\infty }}}$

Taking square on both sides,

$\Rightarrow$ $x^2=8+2\left(\sqrt{8+2\sqrt{8+\mathrm{2...}\infty }}\right)$

$\Rightarrow$ $x^2=8+2x$ [ Since, $x=\sqrt{8+2\sqrt{8+2\sqrt{8+\mathrm{2...}\infty }}}$ ]

$\Rightarrow$ $x^2-2x-8=0$

$\Rightarrow$ $x^2-4x+2x-8=0$

$\Rightarrow$ $x(x-4)+2(x-4)=0$

$\Rightarrow$ $(x-4)(x+2)=0$

$\therefore$ $x=4$ and $x=-2$

$\therefore$ $\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8}}}}}=4$