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
$$
\left\{
\begin{array}{ccll}
x_{2k+1}&=&x_{2k} +x_{2k-1}&\quad (k\ge 1)\\
x_{2k}&=&x_{2k-1} + 2x_{2k-2}&\quad (k\ge 2)
\end{array}
\right.
$$
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 $F_{n+1}=F_{n}+F_{n-1}, (n = 2, 3, 4, \cdots)$
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 $ax^5+by^5$.
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 $(F_{100}, F_{96})$.
How many different strings of length $10$ which contains only letter $A$ or $B$ contains no two consecutive $A$s are there?
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 $$\sum_{k=0}^{n}(-1)^k2^{2n-2k}\binom{2n-k+1}{k}=n+1$$
Compute the value of $$\sum_{k=0}^{n}\frac{1}{2^k}\binom{n+k}{n}$$
Find the number of ways to divide a convex $n$-sided polygon into $(n-2)$ triangles using non-intersecting diagonals.
Find the total number of sequences of length $n$ containing only letters $A$ and $B$ such that no two $A$s are next to each other. For example, for $n = 2$, there are $3$ possible sequences: $AB$, $BA$, and $BB$.
It is well-known that the solution to the Fibonacci sequence is
$$F_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)$$
Show that
$$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2}$$