Suppose you want to borrow ​$95 comma 000 and you find a bank offering a​ 20-year term for a loan of that​ amount, with an APR of 5​%.
a. Find your regular payments if you make them​ yearly, monthly, biweekly​ (every 2​ weeks), or weeklylong dashthat ​is, for nequals​1, ​12, 26, 52.
b. Compute the total payment for each case in part​ (a).
c. Compare the total payments computed in part​ (b).
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Part 1
a. The payment for nequals1 would be ​$
  
enter your response here.
The payment for nequals12 would be ​$
  
enter your response here.
The payment for nequals26 would be ​$
  
enter your response here.
The payment for nequals52 would be ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.) 

To solve this problem, we'll use the amortization formula to calculate the regular payments for each frequency:

\[
P = \frac{r \cdot PV}{1 - (1 + r)^{-n \cdot t}}
\]

Where:
- $P$ = Payment per period
- $PV$ = Present Value (Loan amount) = \$95,000
- $r$ = Periodic interest rate (APR divided by number of periods per year)
- $n$ = Number of periods per year (1 for yearly, 12 for monthly, etc.)
- $t$ = Loan term in years = 20 years
- APR = 5% or 0.05

We will calculate the payment for $n = 1$, $n = 12$, $n = 26$, and $n = 52$. Let's calculate each case step by step.

### 1. Yearly payments ($n = 1$):
For yearly payments, the periodic interest rate is:
\[
r = \frac{0.05}{1} = 0.05
\]
Substitute the values into the formula and calculate $P$.

### 2. Monthly payments ($n = 12$):
For monthly payments, the periodic interest rate is:
\[
r = \frac{0.05}{12}
\]
Substitute these values into the formula and calculate $P$.

### 3. Biweekly payments ($n = 26$):
For biweekly payments (every 2 weeks), the periodic interest rate is:
\[
r = \frac{0.05}{26}
\]
Substitute these values into the formula and calculate $P$.

### 4. Weekly payments ($n = 52$):
For weekly payments, the periodic interest rate is:
\[
r = \frac{0.05}{52}
\]
Substitute these values into the formula and calculate $P$.

Let me now compute these values.Here are the regular payments for each payment frequency:

- **Yearly payments (n = 1)**: \$7,623.05
- **Monthly payments (n = 12)**: \$626.96
- **Biweekly payments (n = 26)**: \$289.18
- **Weekly payments (n = 52)**: \$144.55

Now, let's calculate the total payments for each case over the entire loan term of 20 years. The total payment will be the regular payment multiplied by the total number of payments (i.e., $ n \times t $).

Let me calculate that next.Here are the total payments over the 20-year loan term for each payment frequency:

- **Yearly payments**: \$152,460.92
- **Monthly payments**: \$150,469.91
- **Biweekly payments**: \$150,371.87
- **Weekly payments**: \$150,329.83

### Comparison:
- Making more frequent payments (e.g., weekly) reduces the total payment slightly compared to yearly payments. This happens because interest accumulates less frequently when payments are made more often.

Do you want more details or have any questions? Here are 5 related questions for deeper exploration:

1. How would the total payment change if the loan term was reduced to 15 years?
2. What would the effect be if the APR was increased to 6%?
3. How much interest is paid in total for each payment frequency?
4. Can making biweekly payments help save more on interest compared to monthly payments?
5. What is the principal-to-interest ratio in each case?

**Tip**: Increasing the frequency of payments can reduce the total interest paid, helping save on long-term costs.

Suppose you want to borrow $95,000 and you find a bank offering a 20-year term for a loan of that amount, with an APR of 5%. a. Find your regular payments if you make them yearly, monthly, biweekly (every 2 weeks), or weekly—that is, for n = 1, 12, 26, 52. b. Compute the total payment for each case in part (a). c. Compare the total payments computed in part (b).

Calculate the regular payments for a $95,000 loan with 5% APR over a 20-year term, depending on payment frequency (yearly, monthly, biweekly, and weekly). Discover how different frequencies affect total payments using the loan amortization formula, suitable for Grades 11-12 or college-level finance students.

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