There is a recursive algorithm which solves a problem by dividing the problem into $4$ sub$-$

problems each with half of the original problem size. It takes $D(n)=5n$ time to divide

the problem into sub$-$problems and $C(n)=2n^2$ time to combine the solutions from the sub$-$

problems.

$($a$)$ Write down a recurrence $($recursive formula$)$ for the running time of this algorithm.

$($b$)$ Now you are asked to solve the recurrence that you give above using substitution

method.

i$.$ Make a guess for the asymptotic upper bound for your solution.

ii$.$ Claim an hypothesis based on your guess.

iii. Show that your hypothesis is true.