$(8$ pts$)$ Give a recursive definition with initial condition$($s$).$

a$)$ The function $f(n)=(n+1)!,n=0,1,2,$dots $(4$ pts$)$

b$)$ The Fibonacci numbers $1,1,2,3,5,8,13,$dots. starting from $f(0)(4$ pts$)$

$(8$ pts$)$

a$)$ Find $f(2)$ and $f(3)$ if $f(n)=2f(n-1)+5,f(0)=3.$ Show Steps. $(4$ pts$)$

b$)$ Find $f(8)$ if $f(n)=2f(\frac{n}{2})+1,f(1)=2.$ Show Steps. $(4$ pts$)$