Question

If x = $\sqrt{3 + \sqrt{3 + \sqrt{3 + ....to\infty}}}$then x is equal to

• A. A rational number
• B. An irrational number lying between 2 and 3
• C. An integral number
• D. None of these

Answer 1

Answer: (B)

An irrational number lying between 2 and 3

Answer 2

Given x = $\sqrt{3 + \sqrt{3 + \sqrt{3 + ....\infty}}}$

x = $\sqrt{3 + x}$

squaring both side

x² – x – 3 = 0 Ž x = $\frac{1 \pm \sqrt{1 + 4 \times 1 \times 3}}{2}$

x = $\frac{1 \pm \sqrt{13}}{2}$ ($\frac{1 - \sqrt{13}}{2}$ is not possible)

taking +ve sign x » $\frac{1 + 3.6}{2}$ » 2.3

An irrational number b/w 2 and 3