$(10$ marks$)$ Prove the following using mathematical induction. Show each step clearly.

a$.n^3+2n$ is divisible by $3$

b$.1+3+5+$dots$+2n-1=n^2$

$(10$ marks$)$ Solve the following using Master's method. Show each step clearly.

a$.T(n)=3T(\frac{n}{2})+n^2$

b$.T(n)=4T(\frac{n}{2})+n^2$

c$.T(n)=T(\frac{n}{2})+2^n$

d$.T(n)=2T(\frac{n}{4})+n^{0\mathrm{.}51}$

e$.T(n)=4T(\frac{n}{2})+logn$

$(5$ marks$)$ Fill in the blank using most appropriate symbol from big$-$Oh$,$ bigOmega, or big$-$Theta.

a$.n^k=$

$c^n),$ assuming $k1$ and $c>1$ are constants

b$.lo{g}_{2}n=(lo{g}_{8}n)$

c$.n^{3}lgn=$

$(($:$3nlo{g}_{8}n$