A loan of $\$39,999.85$ is to be repaid by payments at the end of each quarter for eight years. Each payment is $2\%$ higher than its predecessor. The loan is made at a nominal rate of discount of $4\%$ payable quarterly. Find the balance just after the $20th$ payment, the amount of interest in the twentieth payment, and the amount of principal in the twentieth payment.
To find the PV, would this set up be correct? We find $i$ by doing $\frac{.04}{1-.04}=.0416$
$39999.85=\frac {x}{(.041666-.02)}(1-\frac {1+.02}{1+.041666}^{32})$
Let be $L=\$\,39,999.85$, $i^{(4)}=4\%$, $n=8$, $g=2\%$. The effective quarterly rate is $i=\frac{i^{(4)}}{4}=1\%$ and the number of quartely payments is $m=4n=32$. We can find the value of the first payment $P$ from the equation $$ L=\frac{P}{1+i}+\frac{P(1+g)}{(1+i)^2}+\cdots+\frac{P(1+g)^{m-1}}{(1+i)^m}=P\frac{1-\left(\frac{1+g}{1+i}\right)^m}{i-g} $$ that is $P=\$\, 1,079.23$. At time $t=20$ we have that the payment is $$ P_{20}=P(1+g)^{20}= \$\,1,572.23 $$ The outstanding debt at $t=19$ is the present value of the payments $P_{20},\,\ldots,\,P_{32}$ $$ D_{19}=\frac{P(1+g)^{19}}{(1+i)}+\cdots+\frac{P(1+g)^{31}}{(1+i)^{13}}=P(1+g)^{19}\times\frac{1-\left(\frac{1+g}{1+i}\right)^{13}}{i-g}=\$\,21,483.55 $$ The interest paid at $t=20$ is $$ I_{20}=i\,D_{19}= \$\,214.84 $$ and the principal is $$K_{20}=P_{20}-I_{20}= \$\,1,357.40$$ Finally the outstandig debt at time $t=20$ is $$ D_{20}=D_{19}-K_{20}= \$\,20,126.16 $$