$8.$ Use the following algorithm to answer the next sub$-$questions.

for i $=1..$n do

for j $=1..$i do

print $$The proof is in the pudding. $..$

$($a$)(4$ points$)$ Give a closed$-$form sum over number of times the phrase in the inner

loop is issued.

$($b$)(4$ points$)$ Give a closed$-$form summation for every value that j takes in the inner

loop