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Proving sum of convergent geometric series

Given that $\sum\limits_{n=0}^{\infty} x^{n}=1$, find $\sum\limits_{n=0}^{\infty} (x^{n}+y^{n})$

We have that $\sum\limits_{n=0}^{\infty} (x^{n}+y^{n})=\sum\limits_{n=0}^{\infty} x^{n}+\sum\limits_{n=0}^{\infty} y^{n}=1+1=2$

From the binomial theorem we get:

$\sum\limits_{n=0}^{\infty} (x^{n}+y^{n})=\sum\limits_{n=0}^{\infty} (x^{n}+x^{n})=1+\sum\limits_{n=0}^{\infty} x^{n}=1+1=2$

I'm not sure how to solve the problem further. Any help would be appreciated. Thanks!

$\sum_{n=0}^{\infty}(x^n+y^n)=\sum_{n=0}^{\infty}x^n+\sum_{n=0}^{\infty}y^n=1+1=2$

is the correct answer to your question.

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