What i thought was a geometric series of the following form:
$$N = \sum_{i=1}^n[N_{i-1} + (T-N_{i-1})P]$$ where $$ lim_{i \to \infty} N = T$$ and $$N_{i=0}=0$$ I find the series to do the following: $$i=1 : N = TP$$ $$i=2 : N = 2TP-TP^2$$ $$i=3 : N = 3TP-3TP^2 + TP^3$$ $$i=4 : N = 4TP-6TP^2+4TP^3-TP^4$$ $$i=5 : N = 5TP-10TP^2+10TP^3-5TP^4+TP^5$$ $$etc$$
I have been unable to find a solution to reduce this series, and am wondering in general about the pattern emerging.
added: I see that this series mimics a converging inverse exponential, approximately: $$N_n=T(1-e^{-Pn})$$
Your sequence $\{N_i\}$ appears to be defined by $$N_{i+1} = N_{i} + (T-N_{i})P.$$ Therefore $$N_{i+1}-T = (N_{i}-T)(1-P) $$
$$N_k-T = (N_{0}-T)(1-P)^k=-T(1-P)^k.$$
We thus have $$N_k =T(1-(1-P)^k)$$ which, of course, produces the binomial pattern of the terms you have found.