Sequential Adder, Reset Signal

In page 314 of the book we have:

In the lecture, the last relation appeared with mod 2^(i+1-r(i)). Is this a mistake in the book?

The issue is how do you define clock cycle zero?
One way is to define it in an absolute manner (as in the lecture).
Another way, is to say that that if the reset is 1, then we declare the clock cycle to be zero.

I understand we can choose when to set t=0, but how can both of these be true when the LHS and everything else is the same?:
From the book (as in the original post):

From the lecture ( with with mod 2^(i+1-r(i)) ):

Bumping this to get a reply