Modern Appliances Offers A One-year Monthly Installment Plan For A Refrigerator. The Payment For The First Month Is $\$75$, And Then It Increases By $6\%$ Each Month For The Rest Of The Year. Which Expression Can Be Used To Find The

Introduction

In today's world, buying modern appliances can be a significant investment. However, with the help of installment plans, it has become more accessible for people to purchase these appliances. In this article, we will discuss a one-year monthly installment plan for a refrigerator offered by Modern Appliances. The payment for the first month is $\$75$, and then it increases by $6\%$ each month for the rest of the year. We will explore the expression that can be used to find the total cost of the refrigerator.

Calculating the Total Cost

To find the total cost of the refrigerator, we need to calculate the cost of each month and add them up. The cost of the first month is $\$75$. The cost of the second month is $\$75$ plus $6\%$ of $\$75$, which is $\$75 + 0.06 \times 75 = \$79.50$. The cost of the third month is $\$79.50$ plus $6\%$ of $\$79.50$, which is $\$79.50 + 0.06 \times 79.50 = \$84.13$. We can see that the cost of each month is increasing by $6\%$.

Finding the Expression

Let's assume that the cost of the first month is $a$, which is $\$75$. The cost of the second month is $a + 0.06a = 1.06a$. The cost of the third month is $1.06a + 0.06(1.06a) = 1.06^2a$. The cost of the fourth month is $1.06^2a + 0.06(1.06^2a) = 1.06^3a$. We can see that the cost of each month is increasing by a factor of $1.06$.

Using Geometric Series

The cost of each month is increasing by a factor of $1.06$. This is an example of a geometric series, where each term is obtained by multiplying the previous term by a fixed constant. The formula for the sum of a geometric series is:

$$S_n = \frac{a(1 - r^n)}{1 - r}$$

where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Applying the Formula

In this case, the first term $a$ is $\$75$, the common ratio $r$ is $1.06$, and the number of terms $n$ is $12$ (since there are $12$ months in a year). Plugging these values into the formula, we get:

$$S_{12} = \frac{75(1 - 1.06^{12})}{1 - 1.06}$$

Simplifying the Expression

To simplify the expression, we can calculate the value of $1.06^{12}$:

$$1.06^{12} = 1.9673$$

Now, we can plug this value back into the expression:

$$S_{12} = \frac{75(1 - 1.9673)}{1 - 1.06}$$

Evaluating the Expression

To evaluate the expression, we can simplify the numerator and denominator:

$$S_{12} = \frac{75(-0.9673)}{-0.06}$$

$$S_{12} = \frac{72.5475}{-0.06}$$

$$S_{12} = -1210.125$$

However, the total cost cannot be negative. This is because the cost of the refrigerator is increasing each month, and the total cost should be a positive number.

Correcting the Error

The error occurs because the formula for the sum of a geometric series assumes that the common ratio $r$ is less than $1$. In this case, the common ratio $r$ is $1.06$, which is greater than $1$. To correct the error, we can use the formula for the sum of a geometric series when the common ratio is greater than $1$:

$$S_n = \frac{a(r^n - 1)}{r - 1}$$

Applying the Correct Formula

In this case, the first term $a$ is $\$75$, the common ratio $r$ is $1.06$, and the number of terms $n$ is $12$. Plugging these values into the formula, we get:

$$S_{12} = \frac{75(1.06^{12} - 1)}{1.06 - 1}$$

Evaluating the Expression

To evaluate the expression, we can simplify the numerator and denominator:

$$S_{12} = \frac{75(1.9673 - 1)}{0.06}$$

$$S_{12} = \frac{75 \times 0.9673}{0.06}$$

$$S_{12} = \frac{72.5475}{0.06}$$

$$S_{12} = 1210.125$$

Conclusion

$$$$$$$$

Q: What is the monthly installment plan for a refrigerator offered by Modern Appliances?

A: The monthly installment plan for a refrigerator offered by Modern Appliances is a one-year plan where the payment for the first month is $\$75$, and then it increases by $6\%$ each month for the rest of the year.

Q: How much will I pay each month for the refrigerator?

A: The cost of each month will increase by $6\%$ of the previous month's cost. For example, the cost of the second month will be $\$75 + 0.06 \times 75 = \$79.50$, the cost of the third month will be $\$79.50 + 0.06 \times 79.50 = \$84.13$, and so on.

Q: How can I calculate the total cost of the refrigerator?

A: To calculate the total cost of the refrigerator, you can use the formula for the sum of a geometric series:

$$S_n = \frac{a(r^n - 1)}{r - 1}$$

where $a$ is the first term ($\$75$), $r$ is the common ratio ($1.06$), and $n$ is the number of terms ($12$).

Q: What is the total cost of the refrigerator?

A: The total cost of the refrigerator is $\$1210.13$.

Q: Can I pay more than the minimum payment each month?

A: Yes, you can pay more than the minimum payment each month. However, please note that paying more than the minimum payment will not affect the total cost of the refrigerator.

Q: Can I pay less than the minimum payment each month?

A: No, you cannot pay less than the minimum payment each month. If you pay less than the minimum payment, you will be charged a late fee and your credit score may be affected.

Q: What happens if I miss a payment?

A: If you miss a payment, you will be charged a late fee and your credit score may be affected. You may also be subject to additional fees and penalties.

Q: Can I cancel my installment plan?

A: Yes, you can cancel your installment plan at any time. However, please note that you may be subject to a cancellation fee and you will still be responsible for paying the remaining balance.

Q: How can I contact Modern Appliances for more information?

A: You can contact Modern Appliances by phone, email, or in-person at one of their locations. They will be happy to answer any questions you may have and provide more information about the monthly installment plan for a refrigerator.

Conclusion

In this article, we answered some frequently asked questions about the monthly installment plan for a refrigerator offered by Modern Appliances. We hope this information has been helpful in understanding the plan and how it works. If you have any further questions, please don't hesitate to contact Modern Appliances.