Let's say we have exponential random variables . Now consider the following: .
What type of distribution would the sum of exponential random variables, each with mean have?
For starters, you might want to read through this. The continuous stuff is in 7.2. This might also shed light on my reply to your hyperexponential question (if more light is needed .....)
So work it for . Then use that result to get . Then use that result to get . etc. For a result when the random variables are discrete, read through this.