Question $1.(40$ pts$.)$ Consider the loop in sumN$().$ sumN$()$ requires n $>0$ && n $\%2!=0($i$.$e$.,$ n is an odd positive integer$)$ and

returns the sum of odd integers $1$ through n$.$ For example, sumN$(5)=9.$

$//$ precondition: n $>0$ n $\%2=0$

int sumN$($int n$)\{$

int k $=3$;

int sum $=1$;

$//$ LI: sum $=1+3+...+($k $-2)$ k $\%2=03<=$ k $<=($n $+2)$

while $($k $<=$ n$)\{$

sum $=$ sum $+$ k;

k $=$ k $+2$;

$\}$

return sum;

$\}$

$//$ postcondition: sum $=1+3+...+$ n

Given the loop invariant sum $=1+3+...+($k $2)$ k$\%2=03<=$ k $<=($n $+2),$ show that the loop invariant is true for the base

case before the loop executes. $(10$ pts$.)$

Use induction to show that the loop invariant holds for the general case. That is$,$ assume it holds after some iteration m

and show that it holds after iteration m $+1.(20$ pts$.)$

Show that at exit, the loop invariant and the exit condition imply the postcondition. $(10$ pts$.)$