$[$Substitution Method$]$: The first approach for solving a recurrence is the substitution method,

consisting of two steps:

$($a$)$ Begin by guessing the form of the solution, using symbolic constants.

$($b$)$ Use mathematical induction to demonstrate the verify your guess and find the constants.

Use the substitution method to verify the asymptotic solutions for the following recurrences, or

provide a justification for why the suggested solution is incorrect. Note that the Master Theorem

does not apply in this case.

T$($n$)=$T$($n$-1)+$n$^(2)$ has solution T$($n$)=$O$($n$^(3))$

T$($n$)=2$T$($n$-1)+$n has solution T$($n$)=$O$($n$^(5))$