A loan of $12,500 at 9% is to be repaid with n level payments. if in = 128.04, what is the value of n?

present value of annuity = annual payment * [ 1 - (1+i)^-n ]/i

=>

12500 = 128.04 * [1-(1+9%/12^-n]/9%/12

=>n = 42 payments

Present value of annuity formula yields approximately 42 payments for $12,500, with $128.04 monthly payments at 9% interest.

Let's break it down:

[tex]\[PV = Pmt \times \left[1 - \frac{{(1 + i)^{-n}}}{i}\right]\][/tex]

Where:

[tex]- \(PV\) is the present value of the annuity.\\- \(Pmt\) is the amount of each payment in the annuity.\\- \(i\) is the interest rate per period, expressed as a decimal.\\- \(n\) is the total number of payments.[/tex]

Given your values:

- [tex]\(PV\)[/tex] is $12,500.

- [tex]\(Pmt\)[/tex] is $128.04.

- [tex]\(i\)[/tex] is the monthly interest rate, which is [tex]\(9\% / 12 = 0.09 / 12\)[/tex] since the annual rate is divided by 12 for monthly payments.

- [tex]\(n\)[/tex] is what we're solving for.

Substituting these values into the formula, we have:

[tex]\[12,500 = 128.04 \times \left[1 - \frac{{(1 + 0.09/12)^{-n}}}{0.09/12}\right]\][/tex]

Now let's solve for [tex]\(n\):[/tex]

[tex]\[12,500 = 128.04 \times \left[1 - \frac{{(1 + 0.0075)^{-n}}}{0.0075}\right]\]\[1 - \frac{{(1 + 0.0075)^{-n}}}{0.0075} = \frac{{12,500}}{{128.04}}\]\[\frac{{(1 + 0.0075)^{-n}}}{0.0075} = 1 - \frac{{12,500}}{{128.04}}\]\[ (1 + 0.0075)^{-n} = 0.0075 \times \left(1 - \frac{{12,500}}{{128.04}}\right)\]\[ (1 + 0.0075)^{-n} = 0.0075 \times \left(1 - \frac{{12,500}}{{128.04}}\right)\]\[ (1 + 0.0075)^{-n} = 0.9920\]Taking the natural logarithm of both sides:\[ -n \ln(1 + 0.0075) = \ln(0.9920)\][/tex]

Using a calculator:

n ≈ [tex]\frac{{-0.008025}}{{-0.007472}}\][/tex]

n ≈ [tex]41.961\][/tex]

So, rounding up, we get , n ≈ 42

Therefore, it would take approximately 42 payments to reach a present value of $12,500 given the annuity with an annual interest rate of 9% compounded monthly and a payment of $128.04 per month.

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