Relation between geometric series sums

For $|r|<1$, let $S_p=1+r^p+r^{2p}+\dots$ and $s_p=1-r^p+r^{2p}-\dots$ denote the sums of the infinite geometric series. Prove that $S_p+s_p=2S_{2p}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. V. Geometrical Progression, Examples V.a, No. 26