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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 $\frac{n(n+1)}{2}+1$.

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 $$\frac{1}{2}\cdot\frac{3}{4}\cdots\frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$$

Find the value of $x^3+x^2y+xy^2+y^3$ if $x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$.

Simplify $$\sqrt{10+4\sqrt{3-2\sqrt{2}}}$$

Let $\sqrt{1+\sqrt{21+12\sqrt{3}}}=\sqrt{a}+\sqrt{b}$. Find $a+b$.

Let $a\ge 0, n\ge 0,$ and $m > 0$. Show that $\sqrt{a+m}+\sqrt{a+m+n} > \sqrt{a} + \sqrt{a+2m+n}$.

Simplify $(\sqrt{10}+\sqrt{11}+\sqrt{12})(\sqrt{10}+\sqrt{11}-\sqrt{12})(\sqrt{10}-\sqrt{11}+\sqrt{12})(-\sqrt{10}+\sqrt{11}+\sqrt{12})$

Simplify $(\sqrt[3]{3}+\sqrt[3]{2})(\sqrt[3]{9}-\sqrt[3]{6}+\sqrt[3]{4})$

Simplify $\sqrt{1 + 1995\sqrt{4 + 1995 \cdot 1999}}$.

Evaluate $\sqrt{5+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+...}}}}$

Simplify $\sqrt{1-\frac{\sqrt{3}}{2}}$

Simplify $$\frac{1}{2+\frac{1}{2+\cdots}}$$

Simplify $$(\sqrt{2})^{(\sqrt{2})^{(\sqrt{2})^{\cdots}}}$$

Simplify $$2^{\sqrt{2^{\sqrt{2^{\sqrt{2}^{\cdots}}}}}}$$