Problem $1(10$ points$)$: Prove that $n^3-91n^2-7n-14=(n^3).$ Your answer must clearly specify the constants $c$ and $n_0.$

Problem $2(10$ points$)$: Let $g(n)=27n^2+18n$ and $f(n)=0\mathrm{.}5n^2-100.$ Find positive constants $n_0,c_1$ and $c_2$ such that $c_{1}f(n)g(n)c_{2}f(n)$ for all $n{n}_0.$ For full credit, you must show and explain how you derived the constants.

Problem $3$: $(25$ points$)$: For the following pairs of functions $g(n)$ and $f(n),$ justify whether $g(n)$ is $O,o,,$ or $$ of $f(n)$ using rigorous arguments. Just stating the answer will result in $0$ points.

$g(n)=\sqrt[\phantom{2}]{n}$ and $f(n)=(lo{g}_{2}n{)}^2$

$g(n)=n!$ and $f(n)=n^n.$

Problem $4$: $(25$ points$)$: Provide a $O($i$.$e$.,$ Big O$)$ estimate for the code segments below. For full marks, explain how you derived your estimates.

$1.$

$(1)$ Sum $=0$;

$(2)$ for $=0$ Sum $++$; $n=2^{r}rT(n)nT(n)$