Fv Pmt 1 I N-1 I Calculator Solve for N

What is FV = PMT × (1 + i)^n / i?

The formula FV = PMT × (1 + i)^n / i is used to calculate the future value of a series of equal payments (PMT) made at the end of each period, with each payment earning compound interest at a rate of i per period. The variable n represents the number of periods.

This formula is commonly used in finance to calculate the future value of annuities, loans, and other financial instruments where regular payments are made over time with compound interest.

How to Solve for n

To solve for n in the formula FV = PMT × (1 + i)^n / i, you can rearrange the equation using logarithms. Here are the steps:

1. Start with the original formula: FV = PMT × (1 + i)^n / i
2. Multiply both sides by i: FV × i = PMT × (1 + i)^n
3. Divide both sides by PMT: (FV × i) / PMT = (1 + i)^n
4. Take the natural logarithm of both sides: ln((FV × i) / PMT) = ln((1 + i)^n)
5. Use the logarithm power rule: ln((FV × i) / PMT) = n × ln(1 + i)
6. Solve for n: n = ln((FV × i) / PMT) / ln(1 + i)

This logarithmic approach allows you to solve for the number of periods (n) when you know the future value (FV), periodic payment (PMT), and interest rate (i).

Example Calculation

Let's work through an example to illustrate how to solve for n using the formula.

Example Problem

You want to know how many years it will take for a monthly payment of $200 to grow to $50,000 in a savings account that earns 6% annual interest compounded monthly.

Step 1: Identify the Variables

• FV = $50,000
• PMT = $200 (monthly payment)
• i = 6% annual / 12 months = 0.5% or 0.005 (monthly interest rate)

Step 2: Plug into the Formula

Using the rearranged formula: n = ln((FV × i) / PMT) / ln(1 + i)

Step 3: Calculate the Numerator

Numerator = ln((50,000 × 0.005) / 200) = ln((250) / 200) = ln(1.25)

Step 4: Calculate the Denominator

Denominator = ln(1 + 0.005) = ln(1.005)

Step 5: Divide and Solve for n

n = ln(1.25) / ln(1.005) ≈ 1.2218 / 0.00498 ≈ 245.3 months

Step 6: Convert to Years

245.3 months ÷ 12 ≈ 20.44 years

This example demonstrates how the formula can be used to determine the number of periods required to reach a specific future value with regular payments and compound interest.

Common Mistakes

When solving for n in the FV = PMT × (1 + i)^n / i formula, there are several common mistakes to avoid:

1. Incorrect Interest Rate: Ensure the interest rate (i) is expressed as a decimal and is for the correct period (e.g., monthly rate for monthly payments).
2. Logarithm Base: Be consistent with the logarithm base used in your calculations. Natural logarithm (ln) is typically used in financial calculations.
3. Payment Timing: Confirm whether payments are made at the beginning or end of each period, as this affects the calculation.
4. Rounding Errors: Be careful with rounding during intermediate steps, as this can affect the final result.
5. Unit Consistency: Ensure all values (FV, PMT, i) are in consistent units (e.g., dollars, months) to avoid errors.

By being aware of these common mistakes, you can ensure accurate calculations when solving for n in the FV = PMT × (1 + i)^n / i formula.