FORMULABANK

To find the number of periods when you know the initial value, final value, and the percentage growth rate per period.

The formula is as follows:

$n=\frac{\ln \left(\frac{F_n}{F_0}\right)}{\ln (1+r)}$

• F_n = Final capital / final value
• F₀ = Initial capital / initial value
• r = Interest rate / growth rate in % per period
• n = Number of periods

Example:

A bank account with an interest rate of 2% per annum has grown from 500 DKK to 1000 DKK. Over how many years has this occurred?

Calculation Example:

Using the formula:

$n=\frac{\ln \left(\frac{1000}{500}\right)}{\ln (1+0.02)}\approx \mathrm{=35.00279}$

Alternatively:

$1000=500\cdot (\mathrm{(1}+\mathrm{2\%)})$ⁿ

The equation is solved for n using the CAS tool WordMat:

$n=35.00279$

This means it takes approximately 35 years to grow the account to 1000 DKK.