Fill-In-The-Blank Proof By Induction

Fill in the blanks in the inductive proof below of the following property P(N):
å i = (N-1)(N+6)/2, N>=2
For ease of formatting I will write this as follows:

    Sum(i=4,N+2)i = (N-1)(N+6)/2, N>=2.

Now on with the proof.

Base case: N=2. Plugging in 2 for N, we get

 Sum(i=4,4)i  =  (2-1)(2+6)/2  => 4= 4  (show work!).

Inductive Step: (Show that if P(k) is true then P(k+1) must be true.) That is, show that


   Sum(i=4,k+2)i = (k-1)(k+6)/2    (inductive hypothesis)


 Sum(i=4,k+3)i = (k)(k+7)/2  (just like above but k+1 for k)
Proof of the inductive step: By the inductive hypothesis we know that Sum(i=4,k+2)i = (k-1)(k+6)/2. How can we use this (remember, we get to assume that the inductive hypothesis is true), to show that the k+1 case is true? Consider these points: