Compute $$S_n=\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+\cdots+\frac{n+1}{2^n}$$
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
$$
\left\{
\begin{array}{ccll}
x_{2k+1}&=&x_{2k} +x_{2k-1}&\quad (k\ge 1)\\
x_{2k}&=&x_{2k-1} + 2x_{2k-2}&\quad (k\ge 2)
\end{array}
\right.
$$
Is it possible for a geometric sequence to contain three distinct prime numbers?
Is it possible to construct 12 geometric sequences to contain all the prime between 1 and 100?
Let $S_n$ be the sum of first $n$ terms of an arithmetic sequence. If $S_n=30$ and $S_{2n}=100$, compute $S_{3n}$.
Let $d\ne 0$ be the common difference of an arithmetic sequence $\{a_n\}$, and positive rational number $q < 1$ be the common ratio of a geometric sequence $\{b_n\}$. If $a_1=d$, $b_1=d^2$, and $\frac{a_1^2+a_2^2+a_3^2}{b_1+b_2+b_3}$ is a positive integer, what is the value of $q$?
Let $S_n$ be the sum of the first $n$ terms in geometric sequence $\{a_n\}$. If all $a_n$ are real numbers and $S_{10}=10$, and $S_{30}=70$, compute $S_{40}$.
Expanding $$\Big(\sqrt{x}+\frac{1}{2\sqrt[4]{x}}\Big)^n$$
and arranging all the terms in descending order of $x$'s power, if the coefficients of the first three terms form an arithmetic sequence, how many terms with integer power of $x$ are there?
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.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+f(1-x)=2x-x^4$$
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that the Cauchy equation $$f(x+y)=f(x)+f(y)$$ holds for all $x, q\in\mathbb{Q}$.
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 $ax^5+by^5$.
Find the range of function $y=x+\sqrt{x^2 -3x+2}$.
Solve $$\Big|\frac{1}{\log_{\frac{1}{2}}x+2}\Big|> \frac{3}{2}$$
If for any non-negative real numbers $x$ and $y$, function $f(x)$ satisfies the properties that $f(x)\ge 0$, $f(1)\ne 0$, and $f(x+y^2)=f(x)+2f^2(y)$ , compute the value of $f(2+\sqrt{3})$.
If the minimal and maximum values of function $$f(x)=-\frac{1}{2}x^2 + \frac{13}{2}$$ in the domain $[a, b]$ are $2a$ and $2b$, respectively, determine the values of $a$ and $b$.
Is function $f(x)=\lg(x+\sqrt{x^2+1})$ an odd or even function?
For any real numbers $x$ and $y$, the following holds $$[f(x+y)]^2 = [f(x)]^2 + [f(y)]^2$$
Find the exact form of $f(x)$.
Let $f(x)$ be a polynomial with respect to $x$ and $$f(x+1)+f(x-1)=2x^2-4x$$ Find $f(x)$.
Find the function $f(x)$ such that $f(0)=1$, $f(\frac{\pi}{2})=2$, and for any $x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)\cos y$$
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any $m, n\in\mathbb{N}$, $$f(m+n)=f(m)+f(n)+mn$$
Determine $f(n)$.
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any integer $n \ge 2$, $$f(n)=f(n-1) + 2^{n-1}$$
Determine $f(n)$.