Exercise $2$

Consider the following functions:

int recursive$($int $n)$

$\{$

if $)(1$

return $1$ ;

else

return recursive$(n-1$;

$\}$

a$)[5$pts$]$ What does this function calculate?

b$)[5$pts$]$ Give a recurrence relation $)$ for this function. Include both base and recursive cases$)$

c$)[10$pts$]$ Solve the recurrence relation to get the $(dots)$ running time of the function, in terms of $n.$ Show your work, using the substitution method and the recursion tree.

int recursive int $n)$

$\{$

if $)(1$

return $n$;

sum $=0$;

for $($int $2(\frac{n}{3})+$recursive$2(\frac{n}{3})+$recursve$2(\frac{n}{2})+(dots)ni=0$;$i$

sum$++$

return recursive $2(\frac{n}{3})+$recursive$2(\frac{n}{3})+$recursve$2(\frac{n}{2})+$ sum;