Solve The Future Value Formula, $F V = P M T \frac{(1+i)^n-1}{i}$, For $n$.Choose The Correct Answer Below.A. $n = \frac{F V \cdot I}{\operatorname{PMT}(1+i)} + 1$B. $n = \frac{\ln \left(\frac{FV}{PMT}\right)}{\ln

Introduction

The future value formula, $F V = P M T \frac{(1+i)^n-1}{i}$, is a fundamental concept in finance and mathematics. It is used to calculate the future value of an investment based on the present value, interest rate, and time period. However, in some cases, we may need to solve for the number of periods, $n$, given the future value, present value, interest rate, and time period. In this article, we will explore how to solve the future value formula for $n$.

The Future Value Formula

The future value formula is given by:

$$F V = P M T \frac{(1+i)^n-1}{i}$$

where:

• $FV$ is the future value
• $PMT$ is the present value
• $i$ is the interest rate
• $n$ is the number of periods

Solving for n

To solve for $n$, we can start by rearranging the formula to isolate $n$. We can do this by multiplying both sides of the equation by $i$ and then dividing by $PMT$:

$$F V \cdot i = P M T \frac{(1+i)^n-1}{i} \cdot i$$

$$F V \cdot i = P M T \frac{(1+i)^n-1}{1}$$

$$F V \cdot i = P M T (1+i)^n - P M T$$

Now, we can add $PMT$ to both sides of the equation:

$$F V \cdot i + P M T = P M T (1+i)^n$$

Next, we can divide both sides of the equation by $PMT$:

$$\frac{F V \cdot i + P M T}{P M T} = (1+i)^n$$

$$\frac{F V \cdot i}{P M T} + 1 = (1+i)^n$$

Now, we can take the natural logarithm of both sides of the equation:

$$\ln \left(\frac{F V \cdot i}{P M T} + 1\right) = \ln \left((1+i)^n\right)$$

Using the property of logarithms that $\ln a^b = b \ln a$, we can rewrite the right-hand side of the equation as:

$$\ln \left(\frac{F V \cdot i}{P M T} + 1\right) = n \ln (1+i)$$

Now, we can divide both sides of the equation by $\ln (1+i)$:

$$\frac{\ln \left(\frac{F V \cdot i}{P M T} + 1\right)}{\ln (1+i)} = n$$

Conclusion

In conclusion, we have shown that the future value formula can be solved for $n$ by rearranging the formula, taking the natural logarithm of both sides, and then dividing by $\ln (1+i)$. The correct answer is:

$$n = \frac{\ln \left(\frac{F V \cdot i}{P M T} + 1\right)}{\ln (1+i)}$$

This formula can be used to calculate the number of periods, $n$, given the future value, present value, interest rate, and time period.

Example

Suppose we want to calculate the number of periods, $n$, given the following values:

• $FV = 1000$
• $PMT = 500$
• $i = 0.05$
• $n = ?$

Using the formula we derived earlier, we can plug in the values and solve for $n$:

$$n = \frac{\ln \left(\frac{1000 \cdot 0.05}{500} + 1\right)}{\ln (1+0.05)}$$

$$n = \frac{\ln (1.1)}{\ln (1.05)}$$

$$n = \frac{0.0953101798}{0.0487904574}$$

$$n = 1.95$$

Therefore, the number of periods, $n$, is approximately 1.95.

Discussion

The future value formula is a fundamental concept in finance and mathematics. It is used to calculate the future value of an investment based on the present value, interest rate, and time period. However, in some cases, we may need to solve for the number of periods, $n$, given the future value, present value, interest rate, and time period. In this article, we have shown that the future value formula can be solved for $n$ by rearranging the formula, taking the natural logarithm of both sides, and then dividing by $\ln (1+i)$. The correct answer is:

$$n = \frac{\ln \left(\frac{F V \cdot i}{P M T} + 1\right)}{\ln (1+i)}$$

This formula can be used to calculate the number of periods, $n$, given the future value, present value, interest rate, and time period.

References

• [1] Investopedia. (n.d.). Future Value Formula. Retrieved from https://www.investopedia.com/terms/f/futurevalue.asp
• [2] Khan Academy. (n.d.). Future Value Formula. Retrieved from https://www.khanacademy.org/math/ap-calculus-ab/limits-and-sequences-series/sequences-series-ab/v/future-value-formula

Note

The future value formula is a fundamental concept in finance and mathematics. It is used to calculate the future value of an investment based on the present value, interest rate, and time period. However, in some cases, we may need to solve for the number of periods, $n$, given the future value, present value, interest rate, and time period. In this article, we have shown that the future value formula can be solved for $n$ by rearranging the formula, taking the natural logarithm of both sides, and then dividing by $\ln (1+i)$. The correct answer is:

$$n = \frac{\ln \left(\frac{F V \cdot i}{P M T} + 1\right)}{\ln (1+i)}$$

$$

Introduction

In our previous article, we showed how to solve the future value formula for $n$. However, we understand that some readers may still have questions about the formula and how to apply it. In this article, we will answer some of the most frequently asked questions about solving the future value formula for $n$.

Q: What is the future value formula?

A: The future value formula is a mathematical formula used to calculate the future value of an investment based on the present value, interest rate, and time period. It is given by:

$$F V = P M T \frac{(1+i)^n-1}{i}$$

Q: How do I solve the future value formula for $n$?

A: To solve the future value formula for $n$, you can rearrange the formula to isolate $n$. We showed in our previous article that the correct formula is:

$$n = \frac{\ln \left(\frac{F V \cdot i}{P M T} + 1\right)}{\ln (1+i)}$$

Q: What is the natural logarithm?

A: The natural logarithm is a mathematical function that is used to calculate the logarithm of a number to the base $e$. It is denoted by $\ln x$ and is used in many mathematical formulas, including the future value formula.

Q: How do I calculate the natural logarithm?

A: The natural logarithm can be calculated using a calculator or a computer program. You can also use a mathematical table or a logarithm chart to find the natural logarithm of a number.

Q: What is the interest rate, $i$?

A: The interest rate, $i$, is the rate at which interest is earned on an investment. It is usually expressed as a decimal and is used to calculate the future value of an investment.

Q: How do I calculate the interest rate, $i$?

A: The interest rate, $i$, can be calculated using a variety of methods, including the formula:

$$i = \frac{P M T}{F V}$$

Q: What is the present value, $PMT$?

A: The present value, $PMT$, is the current value of an investment. It is the amount of money that is invested at the beginning of a period.

Q: How do I calculate the present value, $PMT$?

A: The present value, $PMT$, can be calculated using a variety of methods, including the formula:

$$P M T = F V \cdot \frac{i}{(1+i)^n-1}$$

Q: What is the future value, $FV$?

A: The future value, $FV$, is the value of an investment at a future date. It is the amount of money that is earned on an investment over a period of time.

Q: How do I calculate the future value, $FV$?

A: The future value, $FV$, can be calculated using the future value formula:

$$F V = P M T \frac{(1+i)^n-1}{i}$$

Conclusion

In this article, we have answered some of the most frequently asked questions about solving the future value formula for $n$. We hope that this article has been helpful in clarifying any confusion about the formula and how to apply it. If you have any further questions, please don't hesitate to contact us.

References

• [1] Investopedia. (n.d.). Future Value Formula. Retrieved from https://www.investopedia.com/terms/f/futurevalue.asp
• [2] Khan Academy. (n.d.). Future Value Formula. Retrieved from https://www.khanacademy.org/math/ap-calculus-ab/limits-and-sequences-series/sequences-series-ab/v/future-value-formula

Note

The future value formula is a fundamental concept in finance and mathematics. It is used to calculate the future value of an investment based on the present value, interest rate, and time period. However, in some cases, we may need to solve for the number of periods, $n$, given the future value, present value, interest rate, and time period. In this article, we have shown that the future value formula can be solved for $n$ by rearranging the formula, taking the natural logarithm of both sides, and then dividing by $\ln (1+i)$. The correct answer is:

$$n = \frac{\ln \left(\frac{F V \cdot i}{P M T} + 1\right)}{\ln (1+i)}$$

This formula can be used to calculate the number of periods, $n$, given the future value, present value, interest rate, and time period.