Question

Find the value of
$\sqrt{6+\sqrt{6+\sqrt{6+}\sqrt{6+........∞}}}$

• A. 2
• B. 6
• C. 9
• D. 3

Answer 1

Answer:

D. 3

Let

$x=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$

Since the expression continues infinitely, the part inside the square root is also (x). Therefore,

$x=\sqrt{6+x}$

Squaring both sides:

$x^2=6+x$
$x^2-x-6=0$

Factorizing:

$(x-3)(x+2)=0$

So,

$x=3 \quad \text{or} \quad x=-2$

Since a square root is non-negative, we take

$\boxed{x=3}.$

Therefore,

$\boxed{\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}=3}.$