$4.(5\mathrm{.}4\mathrm{.}8)$ Suppose that you are given a positive integer n$.($a$)$ Give pseudocode for a recursive algorithm that computes the sum of the integers between $1$ and n$.$ For example, given n $=4,$ your algorithm should return $4$ i$=1$i $=1+2+3+4=10($b$)$ State a lemma establishing the correctness of your algorithm. $($c$)$ Prove that your algorithm is correct. $[$Note: Known facts about this summation may be relevant to your proof, however, you are not asked to prove them. Rather, you are asked to prove that your algorithm correctly computes the summation.$]$