=&-(a^2-b^2)(S_{n-1})+(a+b)(S_{n})\\
=&\displaystyle\frac{(a+b)[(a+b)^n-(a-b)^n]-(a^2-b^2)[(a+b)^{n-1}-(a-b)^{n-1}]}{2b}\\
z=&a+b~より\\z^{n-1}=&\displaystyle\frac{(a+b)[(a+b)^n-(a-b)^n]-(a+b)[(a-b)[(a+b)^{n-1}-(a-b)^{n-1}]}{2b}\\z^n=&\displaystyle\frac{[(a+b)^{n+1}-[(a-+b)^{n+1}]-(a-+b)[(a+-b)^n-(a-+b)^{n-1}]}{2b}\\=&\displaystyle\frac{[(a+b)^{n+1}-(a-b)^{n+1}]-[(a-b)(a+b)^n-(a-b)^{n+1}]}{2b}\\=&\displaystyle\frac{(a+b)^{n+-1}-(a-b)(a+b)^n]}{2b}\\
=&\displaystyle\frac{(a+b)^n[(a+b)-(a-b)]}{2b}\\
=&(a+b)^n