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This is a Yule process, a pure birth process with state-dependent birth rate ${\lambda }_n=\lambda n$. Probability flows into state $n$ from $n-1$ and leaves it for $n+1$, so the master equation is

A simple birth process with parameter $\lambda$ starting from one individual is the Yule process: when its population is $n$, it waits an exponential time of rate $n\lambda$ and then moves to $n+1$.

For $f(n)=n$, the Markov jump-process generator satisfiesTherefore $\mu (t)=\mathbb{E}{X}_t$ obeysIt follows, as recorded by the mean population of a Yule process, that

Let $E_n$ be the holding time in state $n$. The variables are independent with $E_n\sim \mathrm{Exp}(n\lambda )$, and the explosion time would beFor $s>0$,This product is zero because
${\sum }_n\log (1+s/(n\lambda ))=\infty$. If $T_{\infty }$ were finite with positive probability, the nonnegative variable $e^{-s{T}_{\infty }}$ would have positive expectation. Therefore $T_{\infty }=\infty$ almost surely, and the Yule process is nonexplosive.