$1.(5$ pts$)$ Prove

h n$($n$+1)$

$2$

i$2$

$$ n$2($n$2+1)$

$4+98$ in O$($n$3).$

$2.(5$ pts$)$ Prove or disprove $22$n

in O$(22$n$).$

$3.(5$ pts$)$ Prove that any function that is in O$($log$2$ n$)$ is also in O$($log$3$ n$).$

$4.(5$ pts$)$ Prove that if f$1($n$)$ in O$($g$1($n$))$ and f$2($n$)$ in O$($g$2($n$)),$ then

f$1($n$)+$ f$2($n$)$ in O$($g$1($n$)+$ g$2($n$))$