A loan of ${$1000}$ is being repaid with annual payments over 10 years. The size of the payment in the first five years is ${$ k}$. It is found that the payments in the last five years are five times the payments in the first five years. Let the annual rate $r=0.08$, calculate
(i) The value of $K$
(ii)The outstanding loan balance after making the $4th$ payment
(iii) The outstanding loan balance after making the $5th$, hence the principal amortised in the $5th$ payment.
$$ \begin{array}{c|c|c} t=0 & ...t=5 & t=6 ... t=10 \\ \hline \\ K... &K & (5K)... \\ \end{array} $$
(i) In order to find $K$:
$L=K a_{5|0.08}+(5K)(a_{5|0.08}).(1+i)^{-5}$
$1000=K[\frac{1-1.08^{-5}}{0.08}]+5K[\frac{1-1.08^{-5}}{0.08}](1.08)^{-5} \rightarrow K=56.88$
(ii)Using the retrospective method to find $B_4^r$
$B^r_4=1000(1.08)^4-56.88[\frac{1.08^4-1}{0.08}]=1104.18$
(iii)I have tried some different thing. I doubt it works.
$B^5_r=1104.18-56.88[\frac{1.08-1}{0.08}]$=1047.30
My answers are absurd. I get to play more than the loan itself.