# 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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