Please answer the following questions for $T(n).$

$T(n)=4+6+8+$cdots$+2n$

$($a$)$ With the help of the following summation formula of arithmetic series, simplify $T(n).$

Arithmetic series:

${\displaystyle \underset{k=1}{\overset{n}{}}}k=1+2+$dots$+n=\frac{n(n+1)}{2}$

Hint:

$T(n)=4+6+8+$cdots$+2n=2\{2+3+4+$cdots$+n\}$

$($b$)$ Use the induction method to prove the result you get for question $($a$)$ is correct for any $n2.$

$($c$)$ Based on the result you get for question $($a$),$ obtain correct $$ notation for $T(n).$