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$).$