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Let real numbers $a, b, c, d$ satisfy $$ \left\{ \begin{array}{ccl} ax+by&=3\\ ax^2+by^2&=7\\ ax^3+by^3&=16\\ ax^4 + by^4 &=42 \end{array} \right. $$ Find $ax^5+by^5$.

If sequence $\{a_n\}$ has no zero term and satisfies that, for any $n\in\mathbb{N}$, $$(a_1+a_2+\cdots+a_n)^2=a_1^3+a_2^3+\cdots+a_n^3$$ - Find all qualifying sequences $\{a_1, a_2, a_3\}$ when $n=3$. - Is there an infinite sequence $\{a_n\}$ such that $a_{2013}=-2012$? If yes, give its general formula of $a_n$. If not, explain.

We define the Fibonaccie numbers by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n}$. Find the greatest common divisor $(F_{100}, F_{99})$, and $(F_{100}, F_{96})$.

Let $\{a_n\}$ be a sequence defined as $a_1=1$ and $a_n=\frac{a_{n-1}}{1+a_{n-1}}$ when $n\ge 2$. Find the general formula of $a_n$.

For each integer $a_0 >$ 1, define the sequence $a_0, a_1, a_2, \cdots$ by: $$ a_{n+1} = \left\{ \begin{array}{ll} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}\\ a_n + 3 & \text{otherwise} \end{array} \right. $$ For all $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.

Show that $$x+n=\sqrt{n^2 + x\sqrt{n^2+(x+n)\sqrt{n^2+(x+2n)\sqrt{\cdots}}}}$$

Find the value of $$\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$

Let $\{x_n\}$ and $\{y_n\}$ be two real number sequences which are defined as follow: $$x_1=y_1=\sqrt{3},\quad x_{n+1}=x_n +\sqrt{1+x_n^2},\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$ for all $n\ge 1$. Prove that $2 < x_ny_n < 3$ for all $n>1$.

John uses the equation method to evaluate the following expression:$$S=1-1+1-1+1-\cdots$$ and get $$S=1-S \implies \boxed{S=\frac{1}{2}}$$ However, $S$ clearly cannot be a fraction. Can you point out what is wrong here?

The Fibonacci sequence $(F_n)_{n\ge 0}$ is defined by the recurrence relation $F_{n+2}=F_{n+1}+F_{n}$ with $F_{0}=0$ and $F_{1}=1$. Prove that for any $m$, $n \in \mathbb{N}$, we have $$F_{m+n+1}=F_{m+1}F_{n+1}+F_{m}F_{n}.$$ Deduce from here that $F_{2n+1}=F^2_{n+1}+F^2_{n}$ for any $n \in \mathbb{N}$

Find all numbers $n \ge 3$ for which there exists real numbers $a_1, a_2, ..., a_{n+2}$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and\[a_{i}a_{i+1} + 1 = a_{i+2}\]for $i = 1, 2, ..., n.$

Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4$($a_{n-1}$ $+$ $a_{n-2}$ $+$ $a_{n-3}$) is divided by $11$. Find $a_{2018}$ • $a_{2020}$ • $a_{2022}$.

Compute: $1\times 2\times 3 + 2\times 3\times 4 + \cdots + 18\times 19\times 20$.

Compute: $\frac{1}{1\times 2\times 3} + \frac{1}{2\times 3\times 4} + \cdots + \frac{1}{2016\times 2017\times 2018}$

Compute $\frac{1}{1\times 2} + \frac{1}{2\times 3} + \cdots + \frac{1}{2017\times 2018}$

Compute $1\times 2 + 2\times 3 + \cdots + 19\times 20$

Let sequence $\{x_n\}$ satisfy the relation $x_{n+2}=x_{n+1}+2x_n$ for $n\ge 1$ where $x_1=1$ and $x_2=3$.

Let sequence $\{y_n\}$ satisfy the relation $y_{n+2}=2y_{n+1}+3y_n$ for $n\ge 1$ where $y_1=7$ and $y_2=17$.

Show that these two sequences do not share any common term.

Let $a$, $b$, and $x_0$ all be positive integers. Sequence $\{x_n\}$ is defined as $x_{n+1}=ax_n + b$ where $n \ge 1$. Show that $x_1$, $x_2$, $\cdots$ cannot be all prime.

Let sequence $\{a_n\}$ be $a_n=2^n + 3^n + 6^n - 1$ where $n\ge 1$. Find the sum of all positive integers which are co-prime to all the $a_n$.

How many different strings of length $10$ which contains only letter $A$ or $B$ contains no two consecutive $A$s are there?

Let $n$ and $k$ be two positive integers. Show that $$\frac{1}{\binom{n}{k}}=\frac{k}{k-1}\left(\frac{1}{\binom{n-1}{k-1}}-\frac{1}{\binom{n}{k-1}}\right)$$

Let $\{a_n\}$ be a geometric sequence whose initial term is $a_1$ and common ratio is $q$. Show that $$a_1\binom{n}{0}-a_2\binom{n}{1}+a_3\binom{n}{2}-a_4\binom{n}{3}+\cdots+(-1)^na_{n+1}\binom{n}{n}=a_1(1-q)^n$$

where $n$ is a positive integer.

Let $n$ be a positive integer and function $\lfloor{x}\rfloor$ return the largest integer not exceeding $x$. Compute the value of $$\sum_{k=0}^{\lfloor{\frac{n}{2}}\rfloor}\binom{n-k}{k}$$

Show that $$\sum_{k=0}^{n}(-1)^k\frac{m}{m+k}\binom{n}{k}=\frac{1}{\binom{m+n}{n}}$$

Show that $$\sum_{k=0}^{n}(-1)^k2^{2n-2k}\binom{2n-k+1}{k}=n+1$$