Find square root upto infinite times

Q1.  Evaluate

$\sqrt{p\sqrt{p\sqrt{p\sqrt{p\cdots\infty}}}}$

Solution :

Let $x=\sqrt{p\sqrt{p\sqrt{p\sqrt{p\cdots\infty}}}}$. Then

$x^2=p\sqrt{p\sqrt{p\sqrt{p\cdots\infty}}}$  (squaring both sides)

$\Rightarrow x^2=px$

$\Rightarrow (x-p)x=0$

$\Rightarrow x-p=0$  (since $x \neq 0$)

$\Rightarrow x=p$

Q2.  Evaluate

$\sqrt{p+ \sqrt{p+ \sqrt{p+ \sqrt{p+\cdots\infty}}}}$

Solution :

Let $x=\sqrt{p+ \sqrt{p+ \sqrt{p+ \sqrt{p+\cdots\infty}}}}$. Then

$x^2=p+ \sqrt{p+ \sqrt{p+ \sqrt{p+\cdots\infty}}}$  (squaring both sides)

$\Rightarrow x^2=p+x$

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

$\Rightarrow x=\frac{-(-1)\pm\sqrt{(-1)^2-4\cdot 1\cdot (-p)}}{2\cdot 1}$

$\Rightarrow x=\frac{1\pm \sqrt{1+4p}}{2}$