The formula for exponential growth under continuous compounding is given by:

$A = P e^{rt}$

In this equation, $A$ represents the total amount of money accumulated after a specified period, including interest. The variable $P$ denotes the principal amount, which is the initial sum of money invested. The constant $e$ is approximately equal to $2.71828$ and serves as the base of natural logarithms. The variable $r$ indicates the annual interest rate expressed as a decimal, while $t$ signifies the duration of the investment or loan in years.

Continuous compounding implies that interest is added to the principal continuously. This approach differs from traditional compounding methods, which apply interest at discrete intervals such as annually, quarterly, or monthly. As a result, continuous compounding leads to a greater total interest accumulation, since interest is continuously calculated and added to the principal.

To illustrate this concept, let’s consider an example. Imagine you invest £1,000 (where $P = 1000$) at an annual interest rate of 5% (which translates to $r = 0.05$) for a period of 3 years (thus $t = 3$). We can use the formula $A = P e^{rt}$ to determine the total amount:

$A = 1000 \times e^{0.05 \times 3}$

First, we will compute the exponent:

$0.05 \times 3 = 0.15$

Next, we find $e^{0.15}$ using a calculator, which gives us approximately $1.1618$. Now, we multiply this result by the principal:

$A = 1000 \times 1.1618 = 1161.80$

Therefore, after 3 years, your investment would grow to approximately £1,161.80 when compounded continuously.