Consider the following algorithm segment. Assume that n is a positive integer.

for i: $=1$ to $2$n

$$s :$=0$

$$for j :$=1$ to i $1$

$$s :$=$ s $+$ j $($i $$ j $+1)$

$$next j

$$r :$=$ s$2$

next i

$($a$)$

What is the actual number of elementary operations $($additions$,$ subtractions, multiplications, divisions, and comparisons$)$ that are performed when the algorithm segment is executed? For simplicity, count only comparisons that occur within if$-$then statements, and ignore those implied by for$-$next loops. Express your answer in terms of n$.($Hint: See Example $11\mathrm{.}3\mathrm{.}3$ and exercises $11\mathrm{.}3\mathrm{.}11$a and $11\mathrm{.}3\mathrm{.}14$a in the "Read It$"$ link.$)$

The number of operations is

$.$

$($b$)$

Apply the theorem on polynomial orders to the expression in part $($a$)$ to find that an order for the algorithm segment is

n $.$