Consider the following recursive algorithm for computing the sum of the first $n$ cubes: $S(n)=1^3+2^3+...+n^3.$

ALGORITHM $S(n)$

$//$Input: A positive integer $n$

$//$Output: The sum of the first $n$ cubes

if $n=1$ return $1$

else return $S(n-1)+n*n*n$

a$)$ Set up and solve a recurrence relation for the number of times the algorithm's basic operation is executed.

b$)$ How does this algorithm compare with the straightforward non$-$recursive algorithm for computing this sum?