$(12$ pts$)$

a$)(5$ pts$)$ Use the Principle of Mathematical Induction to prove that

$4+8+12+16+$cdots$+4n=2n(n+1)$ for all $n1.$ Show all the steps

b$)(4$ pts$)$ Give a recursive definition with initial condition for the following function.

$f(n)=(3n)!,n=0,1,2,$dots

c$)(3$ pts$)$ In a certain lottery game you choose a set of six numbers out of $30$ numbers.

Find the probability that exactly one of your numbers match the six winning numbers.