Practice (TheColoringMethod)

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Evaluate $\sqrt{5+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+...}}}}$

Simplify $\sqrt{1-\frac{\sqrt{3}}{2}}$

Simplify $$\frac{1}{2+\frac{1}{2+\cdots}}$$

Simplify $$(\sqrt{2})^{(\sqrt{2})^{(\sqrt{2})^{\cdots}}}$$

Simplify $$2^{\sqrt{2^{\sqrt{2^{\sqrt{2}^{\cdots}}}}}}$$

Compute $$\frac{1}{\frac{1}{\frac{1}{\cdots}+1+\frac{1}{\cdots}}+1+\frac{1}{\frac{1}{\cdots}+1+\frac{1}{\cdots}}}$$

Use at least two ways to prove $$\sqrt{x\sqrt{x\sqrt{x\sqrt{\cdots}}}}=x$$

Show that $$\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}=\frac{1}{\frac{1}{1+\frac{1}{1+\cdots}}}=\frac{1+\sqrt{5}}{2}$$

Show that, if both converge, $$\sqrt{a+b\sqrt{a+b\sqrt{a+\cdots}}}=b+\frac{a}{b+\frac{a}{b+\cdots}}=\frac{b+\sqrt{b^2+4a}}{2}$$

Compute $$\sqrt{\frac{2}{2^2}+\sqrt{\frac{2}{2^4}+\sqrt{\frac{2}{2^8}+\cdots}}}$$

Compute $$\sqrt{\frac{2}{2^1}+\sqrt{\frac{2}{2^2}+\sqrt{\frac{2}{2^4}+\cdots}}}$$

Compute $$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$

Compute $$\sqrt{6+2\sqrt{7+3\sqrt{8+\cdots}}}$$


Without using a calculator, explain that $$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}}\approx 1$$


Show that $$\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}=\frac{1+\sqrt{1+4n}}{2}$$ and $$\sqrt{n-\sqrt{n-\sqrt{n-\cdots}}}=\frac{-1+\sqrt{1+4n}}{2}$$

Compute $$\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\cdots}}}$$

Simplify $\sqrt{5\sqrt{3}+6\sqrt{2}}$.

Simplify $\sqrt{12+2\sqrt{6}+2\sqrt{14}+2\sqrt{21}}$

Simplify $\sqrt{\sqrt[3]{9}+6\sqrt[3]{3}+9}$

Simplify $\sqrt{\sqrt[3]{5}-\sqrt[3]{4}}$.

Solve $x^2 +6x - 4\sqrt{5}=0$.

Simplify $\sqrt{4+\sqrt[3]{81}+4\sqrt[3]{9}}$

Simplify $\sqrt{6+\sqrt[3]{81}+\sqrt[3]{9}}$.

A sequence satisfies $a_1 = 3, a_2 = 5$, and $a_{n+2} = a_{n+1} - a_n$ for $n \ge 1$. What is the value of $a_{2018}$?


Let $P(x)$ be a monic polynomial of degree 3. (Monic here means that the coefficient of $x^3$ is 1.) Suppose that the remainder when $P(x)$ is divided by $x^2 - 5x+6$ equals 2 times the remainder when $P(x)$ is divided by $x^2 - 5x + 4$. If $P(0) = 100$, what is $P(5)$?