A discrepancy in understanding a formula.

Question

How is $$(z^{(n +1)} - z^{-(n +1)}) = (z - z^{-1})(z^n + z^{n-1}z^{-1} + ........+zz^{-n +1} + z^{-n}),$$ Could anyone explain this formula for me please how it can be proved?

Answer 1

If you distribute the first term of the first factor on the right ($z$), you have $$z^{n+1}+z^{n-1} + z^{n-3}+\cdots + z^{-n+3} + z^{-n+1}\tag{1}$$ If you distribute the second term of the first factor on the right ($-z^{-1}$), you have $$-(z^{n-1}+z^{n-3}+z^{n-5}+\cdots + z^{-n+1} + z^{-n-1})\tag{2}$$ Adding these, everything adds out to zero except for the first term in (1) and the last term in (2), so you get $$z^{n+1}-z^{-n-1}$$ as required.