Hi everyone,

I read this on a book, which says we have a linear difference equation:

p_{k}= 1/2 (p_{k+1}+ p_{K-1}) if 0<k<N,

s.t. boundary conditions: p_{0}= 1, p_{N}= 0.

The solution is also provided however I don't get it, it says:

Put b_{k}= p_{k}- p_{k-1}, to obtain b_{k}= b_{k-1}and hance b_{k}= b_{1}for all k

[How do we get b_{k}= b_{k-1}?]

Thus, p_{k}= b_{1}+ p_{k-1}= 2b_{1}+ p_{k-2}=...= kb_{1}+ p_{0}

Boundary conditions imply p_{0}= 1, b_{1}= -1/N

so we have p_{k}= 1 - (k/N)

At first glance everything looks fine...but when I read it again I don't quite understand how does b_{k}= b_{k-1}? Could anyone explain why it is the case?

Thank you very much for your help