Having trouble understanding the solution for this question.
A borrower is repaying a loan at 5% effective with payments at the end of each year for 12 years, such that the payment at the end of the first year is $200, at the end of the second year is 190 and so forth until the payment at the end of the 10th year is 110.
(i) Find the amount of the loan.
The solution is as follows:
$L = 100*a_{\overline10|} + 10(Da)_{\overline10|}$
Since the payments start at 200, why is it not
$L = 200*a_{\overline10|} + 10(Da)_{\overline10|}$?
To get the result in the first statement, each payment is broken into two pieces: $$ \begin{align*} \$200 &= \$100 + \$100\\ \$190 &= \$100 + \$90\\ &\,\,\vdots\\ \$110 &= \$100 + \$10\\ \end{align*} $$ so the present value is the present value of the constant $\$100$ stream of payments plus the present value of the decreasing stream of payments.
If you break the payments down as $$ \begin{align*} \$200 &= \$200 - \$0\\ \$190 &= \$200 - \$10\\ &\,\,\vdots\\ \$110 &= \$200 - \$90\\ \end{align*} $$ then you could write the present value as $$200\cdot a_{\overline{10}\vert}-10\cdot d \cdot(I\"{a})_{\overline{9}\vert},$$ which is close to your second statement, but not quite the same.