$4.$ Consider the following algorithm.

ALGORITHM Mystery$($n$)$

$//$Input: A nonnegative integer n

S $0$

for i $1$ to n do

S $$ S $+$ i $$ i

return S

a$.$ What does this algorithm compute?

b$.$ What is its basic operation?

c$.$ How many times is the basic operation executed?

d$.$ What is the efficiency class of this algorithm?

e$.$ Suggest an improvement, or a better algorithm altogether, and indicate its

efficiency class. If you cannot do it$,$ try to prove that, in fact, it cannot be

done.