Problem - 3254
Without using a calculator, explain that $$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}}\approx 1$$
This is because we can show that \begin{align} \sqrt{20+\sqrt{20+\sqrt{20+\sqrt{...}}}}&=5\\ \sqrt{20-\sqrt{20-\sqrt{20-\sqrt{...}}}}&=1 \end{align} Therefore, $$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}}\approx 5 - 4 = 1$$ (Evaluating expressions containing infinite nested radicals is discussed in the book Power Calculation ).