Find the constant term in the expansion of $\left(\frac{x}{2}+\frac{1}{x}+\sqrt{2}\right)^5$.
In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?
Let $n$ be a positive integer and the coefficient of the $x^3$ term in the expansion of $(1+\frac{x}{n})^n$ be $\frac{1}{16}$. Find $n$.
Let $\{a_n\}$ be a geometric sequence whose initial term is $a_1$ and common ratio is $q$. Show that $$a_1\binom{n}{0}-a_2\binom{n}{1}+a_3\binom{n}{2}-a_4\binom{n}{3}+\cdots+(-1)^na_{n+1}\binom{n}{n}=a_1(1-q)^n$$
where $n$ is a positive integer.
Find the coefficient of the $x$ term after having expanded $$(x^2+3x+2)^5$$
Let $X$ be the integer part of $\left(3+\sqrt{7}\right)^n$ where $n$ is a positive integer. Show that $X$ must be odd.
Let the integer and decimal part of $(5\sqrt{2}+7)^{2n+1}$ be $I$ and $D$ respectively. Show that $(I+D)D$ is a constant.
Show that the following inequality holds for any positive integer $n$: $$(2n+1)^n \ge (2n)^n + (2n-1)^n$$
Let $a_n=\binom{200}{n}(\sqrt[3]{6})^{200-n}\left(\frac{1}{\sqrt{2}}\right)^n$, where $n=1$, $2$, $\cdots$, $95$. Find the number of integer terms in $\{a_n\}$.