A loan is repaid with $20$ increasing annual installments of $1,2,3,· · · ,20$. The payments begin one year after the loan is made. Find the principal contained in the $10$th payment, if the annual interest rate is 4%.
Correct Answer: $4.88$
My work:
We need to solve $P_{10} = R_{10}-I_{10} = 10-(0.04)B_9$, where $R_t$=payment amt at time t, $I_t$=amt of interest charged at time t, $B_t$ = outstanding loan balance at time t
Using the prospective method, $$ B_9 = 10a_{\bar{10}|} + 1\cdot \frac{a_{\bar{10}|}-10\nu^{10}}{.04} $$ However, plugging result into final equation gives an answer that is around 5.4. What am I doing wrong?