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Find the length of the leading non-repeating block in the decimal expansion of $\frac{2004}{7\times 5^{2003}}$. For example the length of the leading non-repeating block of $\frac{5}{12}=0.41\overline{6}$ is $2$.

Find $x$ satisfying $x=1+\frac{1}{x+\frac{1}{x+\cdots}}$.

Write $\sqrt[3]{2+5\sqrt{3+2\sqrt{2}}}$ in the form of $a+b\sqrt{2}$ where $a$ and $b$ are integers.

Compute $$\sum_{n=1}^{\infty}\frac{2}{n^2 + 4n +3}$$

Compute $$\sum_{k=1}^{\infty}\frac{1}{k^2 + k}$$

Compute the value of $$\sum_{n=1}^{\infty}\frac{2n+1}{n^2(n+1)^2}$$

Let $(x^{2014} + x^{2016}+2)^{2015}=a_0 + a_1x+\cdots+a_nx^2$, Find the value of $$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$

Find the length of the leading non-repeating block in the decimal expansion of $\frac{2017}{3\times 5^{2016}}$. For example, the length of the leading non-repeating block of $\frac{1}{6}=0.1\overline{6}$ is 1.

Let $\alpha_n$ and $\beta_n$ be two roots of equation $x^2+(2n+1)x+n^2=0$ where $n$ is a positive integer. Evaluate the following expression $$\frac{1}{(\alpha_3+1)(\beta_3+1)}+\frac{1}{(\alpha_4+1)(\beta_4+1)}+\cdots+\frac{1}{(\alpha_{20}+1)(\beta_{20}+1)}$$

If $m^2 = m+1, n^2-n=1$ and $m\ne n$, compute $m^7 +n^7$.

The Lucas numbers $L_n$ is defined as $L_0=2$, $L_1=1$, and $L_n=L_{n-1}+L_{n-2}$ for $n\ge 2$. Let $r=0.21347\dots$, whose digits are Lucas numbers. When numbers are multiple digits, they will "overlap", so $r=0.2134830\dots$, NOT $0.213471118\dots$. Express $r$ as a rational number $\frac{q}{p}$ where $p$ and $q$ are relatively prime.

Let $a_1=a_2=1$ and $a_{n}=(a_{n-1}^2+2)/a_{n-2}$ for $n=3, 4, \cdots$. Show that $a_n$ is an integer for $n=3, 4, \cdots$.

Suppose $\alpha$ and $\beta$ be two real roots of $x^2-px+q=0$ where $p$ and $q\ne 0$ are two real numbers. Let sequence $\{a_n\}$ satisfies $a_1=p$, $a_2=p^2-q$, and $a_n=pa_{n-1}-qa_{n-2}$ for $n > 2$.
• Express $a_n$ using $\alpha$ and $\beta$.
• If $p=1$ and $q=\frac{1}{4}$, find the sum of first $n$ terms of $\{a_n\}$.

• Compute $$S_n=\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+\cdots+\frac{n+1}{2^n}$$

In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m − 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?

Suppose sequence $\{a_n\}$ satisfies $a_1=0$, $a_2=1$, $a_3=9$, and $S_n^2S_{n-2}=10S_{n-1}^3$ for $n > 3$ where $S_n$ is the sum of the first $n$ terms of this sequence. Find $a_n$ when $n\ge 3$.

Find an expression for $x_n$ if sequence $\{x_n\}$ satisfies $x_1=2$, $x_2=3$, and $$\left\{ \begin{array}{ccll} x_{2k+1}&=&x_{2k} +x_{2k-1}&\quad (k\ge 1)\\ x_{2k}&=&x_{2k-1} + 2x_{2k-2}&\quad (k\ge 2) \end{array} \right.$$

Is it possible for a geometric sequence to contain three distinct prime numbers?

Is it possible to construct 12 geometric sequences to contain all the prime between 1 and 100?

Let $S_n$ be the sum of first $n$ terms of an arithmetic sequence. If $S_n=30$ and $S_{2n}=100$, compute $S_{3n}$.

Let $d\ne 0$ be the common difference of an arithmetic sequence $\{a_n\}$, and positive rational number $q < 1$ be the common ratio of a geometric sequence $\{b_n\}$. If $a_1=d$, $b_1=d^2$, and $\frac{a_1^2+a_2^2+a_3^2}{b_1+b_2+b_3}$ is a positive integer, what is the value of $q$?

Let $S_n$ be the sum of the first $n$ terms in geometric sequence $\{a_n\}$. If all $a_n$ are real numbers and $S_{10}=10$, and $S_{30}=70$, compute $S_{40}$.

Expanding $$\Big(\sqrt{x}+\frac{1}{2\sqrt[4]{x}}\Big)^n$$ and arranging all the terms in descending order of $x$'s power, if the coefficients of the first three terms form an arithmetic sequence, how many terms with integer power of $x$ are there?

Suppose sequence $\{F_n\}$ is defined as $$F_n=\frac{1}{\sqrt{5}}\Big[\Big(\frac{1+\sqrt{5}}{2}\Big)^n-\Big(\frac{1-\sqrt{5}}{2}\Big)^n\Big]$$ for all $n\in\mathbb{N}$. Let $$S_n=C_n^1\cdot F_1 + C_n^2\cdot F_2+\cdots +C_n^n\cdot F_n.$$ Find all positive integer $n$ such that $S_n$ is divisible by 8.

Solve $\{L_n\}$ which is defined as $F_1=1, F_2=3$ and $F_{n+1}=F_{n}+F_{n-1}, (n = 2, 3, 4, \cdots)$