<s>[INST] Determinant involving recurrence Evaluate
$$\left| A \right| = \left| {\matrix{
   {x + y} & {xy} & 0 &  \cdots  &  \cdots  & 0  \cr 
   1 & {x + y} & {xy} &  \cdots  &  \cdots  & 0  \cr 
   0 & 1 & {x + y} &  \cdots  &  \cdots  & 0  \cr 
    \cdots  &  \cdots  &  \cdots  &  \cdots  &  \cdots  &  \vdots   \cr 
   0 &  \cdots  & 0 & 1 & {x + y} & {xy}  \cr 
   0 &  \cdots  & 0 & 0 & 1 & {x + y}  \cr 
 } } \right|$$
And show that $\det(A) = \frac{x^{n+1}-y^{n+1}}{x-y}$ if $x\ne y$ and $\det(A) = (n+1)x^n$ if $x=y$.
I actually was able to get this recurrence formula:  
$$D_n = (x+y)\cdot D_{n-1} + xy\cdot D_{n-2}$$
I tried to prove it by induciton, but the algebric calculation didn't bring me to the desired result.
 [/INST] Hint: compute $D_1 = x+y, D_2 = x^2 + xy + y^2$.
Then, prove that $D_n = x^n + x^{n-1}y +  \dots + xy^{n-1} +y^n$ via induction.
Eventually, as $(x^n + x^{n-1}y +  \dots + xy^{n-1} +y^n)(x-y) = x^{n+1} - y^{n+1}$, you are done.
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