$1.(20)$ What is the Big Oh function for each of the following functions of n$?$ You answers should be as tight

and as simple as possible. Also use the definitions of the big oh to preve each of your answers. For example,

if t$($n$)=3\mathrm{.}5$n log$3$ n

$5+$ log log n $+100$

n$,$ the answer should be n log$3$ n and one possible proof could be as

follows:

t$($n$)=3\mathrm{.}5$n log$3$ n

$5+$ log log n $+100$

n

$<=3\mathrm{.}5$n$($log n

$5$

$)$

$3+$ n $+100$n

$=3\mathrm{.}5$n$(5$ log n$)$

$3+101$n

$=437\mathrm{.}5$n log$3$ n $+101$n

$<=437\mathrm{.}5$n log$3$

n $+101$n log$3$

n

$=538\mathrm{.}5$n log$3$ n

$=$ O$($n log$3$ n$)$

where n$0=1$ and C $=538\mathrm{.}5.$ All log functions are base $2.$ And if you use log functions in your answers, they

should be base $2$ as well.

a$.$ t$($n$)=$ n $+($n $+1)++($n $+$ n$)$

b$.$ t$($n$)=$ log$5$ n $\backslash$times log$($n

$10)$

c$.$ t$($n$)=27$log$3$ n

d$.$ t$($n$)=1+5+52+53+...+5$n

e$.$ t$($n$)=2$n $+22$n $+2$n$+1$