Question

The sum of 10 terms of the series √2 + √6 + √18 + ⋯

Answer 1

$S_n=\dfrac{a(1-r^n)}{1-r}\\a=\sqrt2\\r=\sqrt3\\n=10\\\\S_{10}=\dfrac{\sqrt2(1-(\sqrt3)^{10})}{1-\sqrt3}\\\\S_{10}=\dfrac{\sqrt2(1-243)}{1-\sqrt3}\\\\S_{10}=\dfrac{-242\sqrt2}{1-\sqrt3}\\\\S_{10}=\dfrac{-242\sqrt2(1+\sqrt3)}{1-3}\\\\S_{10}=\dfrac{-242\sqrt2-242\sqrt6}{-2}\\\\S_{10}=121\sqrt2+121\sqrt6$