If Monthly Payments Are Made For 30 Years, Find The Value For \[$ N \$\] In The Following Future Value Ordinary Annuity Formula:$\[ F V = P\left(\frac{(1+1)^n - 1}{1}\right) \\]A. 360 B. 12 C. 30 D. \[$\frac{30}{12}\$\]

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Introduction

In finance and mathematics, the future value of an ordinary annuity formula is a crucial concept used to calculate the total value of a series of payments made at regular intervals. The formula is widely used in various fields, including investments, mortgages, and retirement planning. In this article, we will explore the future value of an ordinary annuity formula and provide a step-by-step guide on how to calculate the value of { n $}$ when monthly payments are made for 30 years.

The Future Value of an Ordinary Annuity Formula

The future value of an ordinary annuity formula is given by the following equation:

FV=P((1+1)n−11){ F V = P\left(\frac{(1+1)^n - 1}{1}\right) }

Where:

  • FVFV is the future value of the annuity
  • PP is the periodic payment
  • nn is the number of periods
  • ii is the interest rate per period

Calculating the Value of { n $}$

To calculate the value of { n $}$, we need to rearrange the formula to isolate nn. However, the formula is not in a standard form that allows us to easily solve for nn. To make it more manageable, we can start by simplifying the formula.

Simplifying the Formula

Let's simplify the formula by expanding the exponent:

FV=P((1+i)n−11){ F V = P\left(\frac{(1+i)^n - 1}{1}\right) }

FV=P((1+i)n−1){ F V = P\left((1+i)^n - 1\right) }

Now, let's divide both sides by PP:

FVP=(1+i)n−1{ \frac{FV}{P} = (1+i)^n - 1 }

Rearranging the Formula

To isolate nn, we can add 1 to both sides of the equation:

FVP+1=(1+i)n{ \frac{FV}{P} + 1 = (1+i)^n }

Now, let's take the logarithm of both sides. We can use any base, but let's use the natural logarithm (ln) for simplicity:

ln⁡(FVP+1)=ln⁡((1+i)n){ \ln\left(\frac{FV}{P} + 1\right) = \ln\left((1+i)^n\right) }

Using the property of logarithms that allows us to bring the exponent down, we get:

ln⁡(FVP+1)=nln⁡(1+i){ \ln\left(\frac{FV}{P} + 1\right) = n \ln(1+i) }

Solving for { n $}$

Now, let's divide both sides by ln⁡(1+i)\ln(1+i):

n=ln⁡(FVP+1)ln⁡(1+i){ n = \frac{\ln\left(\frac{FV}{P} + 1\right)}{\ln(1+i)} }

This is the formula we need to calculate the value of { n $}$.

Example

Let's say we have an annuity with the following parameters:

  • FVFV = $100,000
  • PP = $1,000
  • ii = 0.05 (5% interest rate per period)
  • nn = ? (we want to find the value of nn)

Plugging in the values, we get:

n=ln⁡(100,0001,000+1)ln⁡(1+0.05){ n = \frac{\ln\left(\frac{100,000}{1,000} + 1\right)}{\ln(1+0.05)} }

n=ln⁡(101)ln⁡(1.05){ n = \frac{\ln(101)}{\ln(1.05)} }

n=4.60520.0488{ n = \frac{4.6052}{0.0488} }

n=94.3{ n = 94.3 }

Conclusion

In this article, we explored the future value of an ordinary annuity formula and provided a step-by-step guide on how to calculate the value of { n $}$ when monthly payments are made for 30 years. We simplified the formula, rearranged it to isolate nn, and solved for nn using logarithms. The formula we derived is:

n=ln⁡(FVP+1)ln⁡(1+i){ n = \frac{\ln\left(\frac{FV}{P} + 1\right)}{\ln(1+i)} }

This formula can be used to calculate the value of nn in various financial scenarios, including investments, mortgages, and retirement planning.

References

  • [1] Investopedia. (2022). Future Value of an Annuity Formula.
  • [2] Khan Academy. (2022). Future Value of an Annuity.
  • [3] Mathway. (2022). Future Value of an Annuity Calculator.

Discussion

What do you think about the future value of an ordinary annuity formula? Have you ever used it in a financial scenario? Share your thoughts and experiences in the comments below!

Related Articles

Q: What is the future value of an ordinary annuity formula?

A: The future value of an ordinary annuity formula is a mathematical equation used to calculate the total value of a series of payments made at regular intervals. It is widely used in finance and accounting to determine the future value of investments, loans, and other financial instruments.

Q: What are the variables in the future value of an ordinary annuity formula?

A: The variables in the future value of an ordinary annuity formula are:

  • FVFV: the future value of the annuity
  • PP: the periodic payment
  • nn: the number of periods
  • ii: the interest rate per period

Q: How do I calculate the value of { n $}$ using the future value of an ordinary annuity formula?

A: To calculate the value of { n $}$, you need to rearrange the formula to isolate nn. This can be done by simplifying the formula, rearranging it to isolate nn, and solving for nn using logarithms. The formula we derived is:

n=ln⁡(FVP+1)ln⁡(1+i){ n = \frac{\ln\left(\frac{FV}{P} + 1\right)}{\ln(1+i)} }

Q: What is the difference between the future value of an ordinary annuity and the future value of a growing annuity?

A: The future value of an ordinary annuity is a formula used to calculate the total value of a series of payments made at regular intervals, with a fixed interest rate. The future value of a growing annuity, on the other hand, is a formula used to calculate the total value of a series of payments made at regular intervals, with an increasing interest rate.

Q: Can I use the future value of an ordinary annuity formula to calculate the value of a loan?

A: Yes, you can use the future value of an ordinary annuity formula to calculate the value of a loan. However, you need to take into account the interest rate and the number of payments.

Q: What are some common applications of the future value of an ordinary annuity formula?

A: Some common applications of the future value of an ordinary annuity formula include:

  • Calculating the future value of investments
  • Determining the value of a loan
  • Planning for retirement
  • Calculating the future value of a series of payments

Q: Can I use a calculator or software to calculate the future value of an ordinary annuity?

A: Yes, you can use a calculator or software to calculate the future value of an ordinary annuity. Many financial calculators and software programs, such as Excel, have built-in functions to calculate the future value of an ordinary annuity.

Q: What are some common mistakes to avoid when using the future value of an ordinary annuity formula?

A: Some common mistakes to avoid when using the future value of an ordinary annuity formula include:

  • Not taking into account the interest rate
  • Not using the correct formula
  • Not considering the number of payments
  • Not using a calculator or software to check the calculation

Conclusion

In this article, we answered some frequently asked questions about the future value of an ordinary annuity formula. We covered topics such as the variables in the formula, how to calculate the value of { n $}$, and common applications of the formula. We also discussed some common mistakes to avoid when using the formula. We hope this article has been helpful in understanding the future value of an ordinary annuity formula.

References

  • [1] Investopedia. (2022). Future Value of an Annuity Formula.
  • [2] Khan Academy. (2022). Future Value of an Annuity.
  • [3] Mathway. (2022). Future Value of an Annuity Calculator.

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