Question

Please solve this question

Answer 1

Answer:

$\sqrt{2}$

Step-by-step explanation:

Given

$n = 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + ...} } }$

Adding 1 to both sides

$n + 1= 2+ \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + ...} } }$

Note that the denominator of the topmost 1 is n+1 only. So,

$n + 1 = 2 + \frac{1}{n + 1}$

${(n + 1)}^{2} - 2(n + 1) - 1 = 0$

${n}^{2} + 2n + 1 - 2n - 2 - 1 = 0$

${n}^{2} - 2 = 0$

${n}^{2} = 2$

$n = \sqrt{2}$