Section 3.3 Determining interest for any given payment
To determine the amount of interest in the kth payment, one can avoid having to compute the entire amortization schedule by computing the present value \(P_{mid}\) of the final n-k payments at the previous time period. This will be at the n-(k-1 Then, compute the interest demanded on that amount one time period later. That is, \(P_{mid} \cdot r \text{.}\)
\begin{gather*}
P_{mid} = R \frac{1-(1+r)^{-(n-(k+1))}}{r}\\
P_{mid} \cdot r = R (1-(1+r)^{k-n-1})
\end{gather*}