Show that $2^n=O(n!)$

Show that $2+4+8+$dots$+2^n$ is $(2^n).$

Show that $5+10+15+$dots$+5n$ is $(n^2)$

Show that $1^3+2^3+3^3+$dots$+n^3$ is $(n^4).$

Show that $n+\frac{n}{2}+\frac{n}{4}+$dots$+\frac{n}{2^k}$ is $(n).($b$)20($c$)200($d$)4004.$ Find $\backslash$Theta $($f$($n$))$ for the algorithm for i$=1$ to $2$n for j$=1$ to n$,$n$^(2)$ x$=$x$+15.$ Find $\backslash$Theta $($f$($n$))$ for the algorithm for i$=1$ to $2$n$,$i$=1,$i$=2$ for j$=1$ to $|\_(\_())($i$+1)/((2)\_(\_()))(||)/($ b$)$ar $($J$)|=1$ to $1$ x$=$i$*$j