Consider the following block of pseudocode that calls a

subroutine DoWork on an array A of length n passing in an index i$.$

$1$ for i $=1$ to n

$2$ for j $=1$ to i

$3$ DoWork$($A$,$ i$)$

Suppose that DoWork$($A$,$ k$)$ needs k $+1$ units of work. Let T $($n$)$ be the total units

of work needed for the above block on an array of size n$.$

$($a$)$ Come up with a summation for the total units of work needed as a function of

n$.$

$($b$)$ Use induction to prove a closed form for your summation.

$($c$)$ Give a tight $($big$-\backslash$Theta $)$ asymptotic bound on T $($n$).$