Distribution of sum of exponential variables with different parameters

you can use the main result here applied to the sequence $\lambda, \lambda/2,...,\lambda/k$, which for the general case states that

$$ h_{X_1,.,X_n}(x) = \left[ \prod_{i=1}^n \lambda_{i} \right] \sum_{j=\ 1}^n \frac{e^{-\lambda_i x}}{\prod_{k \neq j}^n(\lambda_i-\lambda_j)} $$