A given cell can replicate at rate $\lambda$ and die at rate $\mu$. Upon replication, the cell divides into 2.
The question asks: how many cells will be produced by this cell before it dies?
Case 1) I only count the cells which are born directly by this cell, e.g. if it replicates, I don't count the replications of the 2nd cell.
Case 2) I consider the total number of cells generated.
Case 1: With probability $\lambda / \mu$ the cell will replicate before it dies, with probability $(\lambda / \mu)^2$ it will replicate twice and so on using the memoryless property of the exponential. Each time we get an extra cell, hence we have $E(X)=\sum_1^\infty (\lambda /\mu)^i \times i$
Case 2: I don't know how to include all possible divisions of daughter cells. Maybe it's not required?