Induction. Consider the following recurrence definition:

$T(1)=1$

$T(i+1)=T(i)+3i^2+3i+1$

a$.$ Prove by induction the predicate

$)=(i^3$

b$.$ Give the Big Oh$()$ attribute of the following algorithm as a function of $N$ : int $T,i$;$,T=1$;$i=1$;

while $)(N$

$\{T=T+3**i**i+3**i+1$;$i=i+1$;$\}$