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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 $a_{2018}$?

For $n\ge 1$, let $d_n$ denote the length of the line segment connecting the two points where the line $y = x + n + 1$ intersects the parabola $8x^2 = y - \frac{1}{32}$ . Compute the sum $$\sum_{n=1}^{1000}\frac{1}{n\cdot d_n^2}$$

Determine all pairs $(a, b)$ of real numbers such that $10, a, b, ab$ is an arithmetic progression.

Let $$f(r) = \displaystyle\sum_{j=2}^{2008}\frac{1}{j^r} = \frac{1}{2^r}+\frac{1}{3^r}+\cdots+\frac{1}{2016^r}$$ Find $$\sum_{k=2}^{\infty}f(k)$$

Evaluate the infinite sum $\displaystyle\sum_{n=1}^{\infty}\frac{n}{n^4+4}$.

Solve the equation $$\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\cdots+\sqrt{4^{2008}x+3}}}}}-\sqrt{x}=1$$

Find all $x$ such that $\displaystyle\sum_{k=1}^{\infty}kx^k=20$.