A loan of $10,000$ is being repaid with 20 non-level annual payments. The interest rate on the loan is an annual effective rate of$6$% . The loan was originated 4 years ago. Payments of $500$ at the end of the first year, $750$ at the end of the second year, $1000$ at the end of the third year, $1250$ at the end of the forth year have been paid.
Calculate the outstanding balance immediately after the forth payment.
I tried through
Present value of all payments = Amount of Loan
$\frac{500}{1.06} +\frac{750}{1.06^2}+\frac{1000}{1.06^3}+\frac{1250}{1.06^4}+B_{(4)}^r=1000$
Getting some problems here. This equation does not fit in.
After first year, the balance is 10000*1.06-500=10100. After the second year, the balance is 10100*1.06-750=9956. After the third year, the balance is 9956*1.06-1000=9553.36. After the 4th year, 8876.5616. In general, for $P_i$ payments at the end of year $i$, the balance is $$B^r_{(n)}=10000 \times 1.06^n-\sum_{i=1}^n P_i\times1.06^{n-i}$$