Nt+1 = λNt ⇔ ΔNt to t+1 = RNt I'm struggling in my subject that has a component of maths in it, please help.
I need to make $N_{t+1} = \lambda Nt$ become $\Delta N_{t\ \text{to}\ t+1} = RN_t$ showing working.
$N_{t+1}$ is the population size at time $t+1$.
$N_t$ is the population size at $t$.
$\lambda$ is a constant factor by which the population increases per time unit.
Thank you.

In your course or book, something that looks more or less like $\Delta N_{t\;\text{to}\;t+1}$ has been defined as the difference between $N_{t+1}$ and $N_t$, that is, as $N_{t+1}-N_t$. This is the change in population from time $t$ to time $t+1$.  (A more common notation for the same thing is the more compact $\Delta N_t$.)
We have 
$$N_{t+1}=\lambda N_t.$$
It follows that 
$$\Delta N_{t\;\text{to}\;t+1}=N_{t+1}-N_t=\lambda N_t -N_t=(\lambda-1)N_t.$$
Thus 
$\Delta N_{t\;\text{to}\;t+1}  =(\lambda-1)N_t=RN_t$, where $R=\lambda-1$.
Note: Sometimes people have trouble with the fact that $\lambda N_t -N_t=(\lambda-1)N_t$, even though they would have no trouble seeing that, for example, $\lambda N_t-5N_t=(\lambda-5)N_t$.  We need to observe that $N_t=1\cdot N_t\;$:$\:$  If this year's population is $N$, and next year's population is $17N$, the change in population is $17N -N$, which is $16N$. 
