Zipper Has N Links Each Link Has A State In

Zipper has N links: each link has a state in which it is closed with energy 0 and a state in which it is open with energy ε. We require, however, that the zipper can only unzip from the left end, and that the link number s can only open if all links to the left (1, 2 . . . s – 1) are already open.
(a) Show that the partition function can be summed in the form

Z = 1 – exp [–(N + 1)ε/τ] / 1 – exp (–ε/τ)

(b) In the limit ε >> τ, find the average number of open links. The model is very simplified model of the unwinding of two-stranded DNA molecules.

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