Suppose function $f(x)=\frac{1+x}{1-x}$. Evaluate $$f\Big(\frac{1}{2}\Big)\cdot f\Big(\frac{1}{4}\Big)\cdot f\Big(\frac{1}{6}\Big)\cdots f\Big(\frac{1}{2014}\Big)$$
Let the sum of first $n$ terms of arithmetic sequence $\{a_n\}$ be $S_n$, and the sum of first $n$ terms of arithmetic sequence $\{b_n\}$ be $T_n$. If $\frac{S_n}{T_n}=\frac{2n}{3n+7}$, compute the value of $\frac{a_8}{b_6}$.
Suppose every term in the sequence
$$1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, \cdots$$
is either $1$ or $2$. If there are exactly $(2k-1)$ twos between the $k^{th}$ one and the $(k+1)^{th}$ one, find the sum of its first $2014$ terms.
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.
What is the Ptolemy Theorem?
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 $n$ be a positive integer. Show that $$\Big(1+\frac{1}{3}\Big)\Big(1+\frac{1}{3^2}\Big)\cdots\Big(1+\frac{1}{3^n}\Big) < 2$$
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 $a_n > \frac{\pi}{2^{n+2}}$.
Assuming a small packet of mm’s can contain anywhere from $20$ to $40$ mm’s in $6$ different colours. How many different mm packets are possible?
Show that $$\sum_{k=1}^n \binom{n}{k}\binom{n}{k-1}=\binom{2n}{n-1}$$
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 $\frac{1}{1\times 2\times 3}+\frac{1}{2\times 3\times 4}+\cdots + \frac{1}{n\times (n+1)\times (n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$