$1//$ pre: M $>=1$

$2$ j $1$

$3$ S $0$

$4//$ LoopInv:

$5$ while $($j $<=$ M$)$

$6$ S $$ S $+(2*$j $-1)*(2*$j $-1)$

$7$ j $$ j$+1$

$8//$ post: S $==$ M$*(4*$M$*$M $-1)/3$

$($a$)$ State and prove a loop invariant for this program.