Connecting heat engines

Lets consider N identical heat engines connected in series, with heat $Q_{delivered}$ being delivered to the "first" engine. Since we know that:

$\eta _{0}=\frac{ W }{Q_{delivered}}$

and:

$W=Q_{delivered}-Q_{rejected}$

We can easily conclude that for the "first" engine:

$Q_{rejected}=Q_{delivered}(1-\eta_{0})$

For the second engine, the heat rejected by the first one is the heat delivered, which means that for the "second" engine:

$\eta _{0}=\frac{ W_{2} }{Q_{delivered}(1-\eta_{0})}$

and:

$W_{2}=Q_{delivered}(1-\eta_{0})-Q_{rejected_2}$

$Q_{rejected_2}=Q_{delivered}(1-\eta_{0})-\eta_{0}Q_{delivered}(1-\eta_{0})=Q_{delivered}(1-\eta_{0})^2$

It is now easy to conclude (or prove by induction if you wish) that for N identical heat engines, the total rejected heat will be equal to:

$Q_{rejected_N}=Q_{delivered}(1-\eta_{0})^N$

And so, the effective efficiency of such a system will be:

$\eta_{eff}=1-\frac{Q_{rejected_N}}{Q_{delivered}}=1-(1-\eta_{0})^N$

substituting $\eta_{0}=0.3$ lets us easily get to the answer: