First, you should always include the parametrization of the gamma you are using when posting questions. Some parameterize it so that whereas other parametrize it so that . My guess is that this question is assuming the later parametrization, but the way that you worked the problem suggests that you are using the former. I'll assume the latter.
The sum of all our random variables is a Gamma(200, 1/10), which is approximately normal with mean 2000 and variance 20000, i.e. the standard deviation is approximately 141.42. Get a Z value:
(2191.5 - 2000) / 141.42 = 1.35
and calculate P(Z > 1.35) to get your answer.