The sum to 10 terms of the series $ \sqrt{2}+\sqrt{6}+\sqrt{18}+\sqrt{54}+...$ is

The series is sqrt(2), sqrt(6), sqrt(18), sqrt(54). This is a geometric series with a = sqrt(2) and r = sqrt(3). Sum = a(r^n - 1) / (r - 1) = sqrt(2)(sqrt(3)^10 - 1) / (sqrt(3) - 1) = sqrt(2)(243 - 1) / (sqrt(3) - 1) = 242 * sqrt(2) / (sqrt(3) - 1). Rationalizing gives 121 * sqrt(2) * (sqrt(3) + 1) = 121(sqrt(6) + sqrt(2)).