Sum thing

Algebra Level 4

\[\large \begin{cases} x+y=2 \\ xy=4 \\ { S }_{ n }={ x }^{ n }+{ y }^{ n } \end{cases}\] The above equations are given, where \(n\) is a positive integer. It can be shown that

\[\large \color{blue}{p} S_n=S_{n+1}+ \color{red}{q} S_{n-1}\]

where \(\color{blue}{p}\) and \(\color{red}{q}\) are positive integers. Find \(\color{blue}{p} \color{red}{q}\).

Bonus questions:

  • Find a similar formula for \(D_n=x^n-y^n\).
  • Find a similar formula which involves \(D_n\), \(S_n\) and \(i=\sqrt{-1}\), given that \(\text{Im}(x)>0\).

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