The Fibonacci numbers are defined by $F_1=1, F_2=1$, and $F_n=F_{n-1} + F_{n-2}$ for $n=3, 4, \cdots$. Find and prove a formula for the sum of the first $n$ Fibonacci numbers, i.e. $F_1 + F_2 + \cdots +F_n$.
Let {$a_n$} be a sequence with $a_1=1$. If for any $n > 1$, $a_n$ equals one plus twice of the sum of all the previous terms, express $a_n$ in terms of $n$.
How many among the first $1000$ Fibonacci numbers are multiples of $11$?
Let $F(1)=1, F(2)=1, F(n+2)= F(n+1)+F(n)$ be the Fibonacci sequence. Prove if $i | j$, then $F(i) | F(j)$. In another word, every $k^{th}$ element is a multiple of $F(k)$.
Find the maximal value of $m^2+n^2$ if $m$ and $n$ are integers between $1$ and $1981$ satisfying $(n^2-mn-m^2)^2=1$.
Find the general formula of the sequence defined as $a_1=6$ and $a_n=\frac{1}{2}a_{n-1}+4$.
There is a sequence with $a(2) = 0$, $a(3) = 1$ and $a(n) = a(\lfloor{\frac{n}{2}}\rfloor)+a(\lceil{\frac{n}{2}}\rceil)$ for $n\ge 4$. Find $a(2014)$.
Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n =\frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?
Suppose that $(u_n)$ is a sequence of real numbers satisfying $u_{n+2}=2u_{n+1}+u_n$, and that $u_3=9$ and $u_6=128$. What is $u_{2015}$?
Let sequence $\{a_n\}$ satisfy $a_0=0, a_1=1$, and $a_n = 2a_{n-1}+a_{n-2}$. Show that $2^k\mid n$ if and only if $2^k\mid a_n$.
Let sequence $g(n)$ satisfy $g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$ where $n\ge 1$. Show that if $n$ is a prime greater than 5, then $n\mid g(n)[g(n)+1]$.
Show that all the terms of the sequence $a_n=\frac{(2+\sqrt{3})^n-(2-\sqrt{3})^n}{2\sqrt{3}}$ are integers, and also find all the $n$ such that $3 \mid a_n$.
Show that all terms of the sequence $a_n=\left(\frac{3+\sqrt{5}}{2}\right)^n+\left(\frac{3-\sqrt{5}}{2}\right)^n -2$ are integers. And when $n$ is even, $a_n$ can be expressed as $5m^2$, when $n$ is odd $a_n$ can be expressed as $m^2$.
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles?
Determine all polynomials such that $P(0) = 0$ and $P(x^2 + 1) = P(x)^2 + 1$.
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?
Show that $1(1!)+2(2!)+3(3!)+\cdots+n(n!)=(n+1)!-1$
Show that $|\sin(nx)|\le n|\sin(x)|$ for any positive integer $n$.
$n$ straight lines are drawn in the plane in such a way that not two of them are parallel and not three of them meet at one point. Show that the number of regions in which these lines divide the plane is $\frac{n(n+1)}{2}+1$.
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 for any $m, n \in\mathbb{N}$, we have $$F_{m+n+1}=F_{m+1}{n+1}+F_mF_n$$
Deduce from here that $F_{2n+1}=F_{n+1}^2 +F_n^2$ for any $n\in\mathbb{N}$.
Show that $$\frac{1}{2}\cdot\frac{3}{4}\cdots\frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$$
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 $a_{2018}$?
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.
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 $\{a_n\}$.