Let $c_1, c_2, c_3, \cdots$ be a series of concentric circles whose radii form a geometric sequence with common ratio as $r$. Suppose the areas of rings which are formed by two adjacent circles are $S_1, S_2, S_3, \cdots$. Which statement below is correct regarding the sequence $\{S_n\}$?
A) It is not a geometric sequence
B) It is a geometric sequence and its common ratio is $r$
C) It is a geometric sequence and its common ratio is $r^2$
D) It is a geometric sequence and its common ratio is $r^2-1$
Given the sequence $\{a_n\}$ satisfies $a_n+a_m=a_{n+m}$ for any positive integers $n$ and $m$. Suppose $a_1=\frac{1}{2013}$. Find the sum of its first $2013$ terms.
Let sequence $\{a_n\}$ satisfy $a_1=2$ and $a_{n+1}=\frac{2(n+2)}{n+1}a_n$ where $n\in \mathbb{Z}^+$. Compute the value of $$\frac{a_{2014}}{a_1+a_2+\cdots+a_{2013}}$$
Let $a_1, a_2,\cdots, a_n > 0, n\ge 2,$ and $a_1+a_2+\cdots+a_n=1$. Prove $$\frac{a_1}{2-a_1} + \frac{a_2}{2-a_2}+\cdots+\frac{a_n}{2-a_n}\ge\frac{n}{2n-1}$$
Suppose all the terms in a geometric sequence $\{a_n\}$ are positive. If $|a_2-a_3|=14$ and $|a_1a_2a_3|=343$, find $a_5$.
Suppose no term in an arithmetic sequence $\{a_n\}$ equals $0$. Let $S_n$ be the sum of its first $n$ terms. If $S_{2n-1} = a_n^2$, find the expression for its $n^{th}$ term $a_n$.
Let $S_n$ be the sum of first $n$ terms in sequence $\{a_n\}$ where $$a_n=\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}$$ Find $\lfloor{S_n}\rfloor$ where the floor function $\lfloor{x}\rfloor$ returns the largest integer not exceeding $x$.
Let $\alpha$ and $\beta$ be the two roots of the equation $x^2 -x - 1=0$. If $$a_n = \frac{\alpha^n - \beta^n}{\alpha -\beta}\quad(n=1, 2, \cdots)$$ Show that
- For any positive integer $n$, it always hold $a_{n+2}=a_{n+1}+a_n$
- Find all positive integers $a, b$ $( a < b )$ satisfying $b\mid a_n-2na^n$ holds for any positive integer $n$
Let $\{a_n\}$ be an increasing geometric sequence satisfying $a_1+a_2=6$ and $a_3+a_4=24$. Let $\{b_n\}$ be another sequence satisfying $b_n=\frac{a_n}{(a_n-1)^2}$. If $T_n$ is the sum of first $n$ terms in $\{b_n\}$, show that for any positive integer $n$, it always holds that $T_n < 3$.
Given a sequence $\{a_n\}$, if $a_n\ne 0$, $a_1=1$, and $3a_na_{n-1}+a_n+a_{n-1}=0$ for any $n\ge 2$, find the general term of $a_n$.
If a sequence $\{a_n\}$ satisfies $a_1=1$ and $a_{n+1}=\frac{1}{16}\big(1+4a_n+\sqrt{1+24a_n}\big)$, find the general term of $a_n$.
Let sequence $\{a_n\}$ satisfy $a_0=1$ and $a_n=\frac{\sqrt{1+a_{n-1}^2}-1}{a_{n-1}}$. Prove $a_n > \frac{\pi}{2^{n+2}}$.
Show that $1+3+6+\cdots+\frac{n(n+1)}{2}=\frac{n(n+1)(n+2)}{6}$.
Show that $1+4+7+\cdots+(3n-2)=\frac{n(3n-1)}{2}$
Show that $1^2 + 3^2 + \cdots + (2n-1)^2=\frac{n(2n-1)(2n+1)}{3}$
Show that $1+5+9+\cdots+(4n-3)=2n^2 -n$
Show that $2+2^3 + 2^5+\cdots+2^{2n-1}=\frac{2(2^{2n}-1)}{3}$.
Show that $\frac{1}{1\times 2\times 3}+\frac{1}{2\times 3\times 4}+\cdots + \frac{1}{n\times (n+1)\times (n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$
Show that $4^3 + 8^3 + 12^3 + \cdots + (4(k+1))^3=16(k+1)^2(k+2)^2$.
Show that $\frac{1}{5^2}+\frac{1}{5^4}+\cdots+\frac{1}{5^{2n}}=\frac{1}{24}(1-\frac{1}{25^n})$.
Show that $2^{-1}+2^{-2}+2^{-3}+\cdots+2^{-n}=1-2^{-n}$.
Show that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{3^3}+\cdots+\frac{n}{2^n}=2-\frac{n+2}{2^n}$
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 $\frac{n(n+1)}{2}+1$.