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Answer :
To determine the monthly payment for a \[tex]$170,000 loan over 20 years with an annual interest rate of 12.6% compounded monthly, we first convert the annual rate into a monthly rate. Since there are 12 months in a year, the monthly interest rate is
$[/tex][tex]$
r = \frac{12.6\%}{12} = \frac{0.126}{12} = 0.0105.
$[/tex][tex]$
Next, the total number of monthly payments over 20 years is
$[/tex][tex]$
n = 20 \times 12 = 240.
$[/tex][tex]$
The formula to calculate the monthly payment for a loan is
$[/tex][tex]$
P = \frac{L \, r \, (1+r)^n}{(1+r)^n - 1},
$[/tex][tex]$
where
\( L \) is the loan principal,
\( r \) is the monthly interest rate, and
\( n \) is the total number of payments.
Substituting the given values into the formula gives
$[/tex][tex]$
P = \frac{170\,000 \times 0.0105 \times (1+0.0105)^{240}}{(1+0.0105)^{240} - 1}.
$[/tex][tex]$
Notice that the expression
$[/tex][tex]$
\frac{170\,000 \times 0.0105 \times (1+0.0105)^{240}}{(1+0.0105)^{240} - 1}
$[/tex][tex]$
matches the expression given in option D (after accounting for a typographical representation of \$[/tex]170,000 in the option).
Thus, the correct answer is option D.
$[/tex][tex]$
r = \frac{12.6\%}{12} = \frac{0.126}{12} = 0.0105.
$[/tex][tex]$
Next, the total number of monthly payments over 20 years is
$[/tex][tex]$
n = 20 \times 12 = 240.
$[/tex][tex]$
The formula to calculate the monthly payment for a loan is
$[/tex][tex]$
P = \frac{L \, r \, (1+r)^n}{(1+r)^n - 1},
$[/tex][tex]$
where
\( L \) is the loan principal,
\( r \) is the monthly interest rate, and
\( n \) is the total number of payments.
Substituting the given values into the formula gives
$[/tex][tex]$
P = \frac{170\,000 \times 0.0105 \times (1+0.0105)^{240}}{(1+0.0105)^{240} - 1}.
$[/tex][tex]$
Notice that the expression
$[/tex][tex]$
\frac{170\,000 \times 0.0105 \times (1+0.0105)^{240}}{(1+0.0105)^{240} - 1}
$[/tex][tex]$
matches the expression given in option D (after accounting for a typographical representation of \$[/tex]170,000 in the option).
Thus, the correct answer is option D.
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