Question 5

$$\sqrt{6+\sqrt{6+\sqrt{6+}}}$$..... is equal to

Solution

Let $$\sqrt{\ 6+\sqrt{\ 6+\sqrt{\ 6+........}}} = x$$

Therefore, $$\sqrt{\ 6+x} = x$$

$$6+x = x^{2}$$

$$x^{2}-x-6 = 0$$

$$x^2-3x+2x-6=0$$

$$x\left(x-3\right)+2\left(x-3\right)=0$$

$$\left(x-3\right)\left(x+2\right)=0$$

$$x=3,\ x=-2$$

Hence, $$\sqrt{\ 6+\sqrt{\ 6+\sqrt{\ 6+........}}}=\ 3$$

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