Exponential Growth and Decay | Understanding the A(t) = P(1 +/- r)^t Formula

A(t)=P(1 +/- r)^t

In the formula A(t) = P(1 +/- r)^t, we are looking at exponential growth or decay

In the formula A(t) = P(1 +/- r)^t, we are looking at exponential growth or decay. Let’s break down the components of this formula:

– A(t) represents the amount or value at time t. This is usually the quantity we want to find or calculate.

– P represents the initial amount or value. It is the starting point for the exponential growth or decay process.

– r represents the rate of growth or decay expressed as a decimal. If the rate is positive, we have exponential growth, and if the rate is negative, we have exponential decay.

– (1 +/- r) is the base of the exponential function. If the rate is positive, we use (1 + r) for growth, and if the rate is negative, we use (1 – r) for decay.

– t represents the time that has passed since the initial amount. It could be in years, months, days, or any unit of time.

To use this formula, you need to know the values for P, r, and t. Plug those values into the formula, and you will find the amount or value at the given time.

If the rate is positive, you add 1 to r, and if the rate is negative, you subtract r from 1 before raising it to the power of t. This is because we are dealing with multiplication in exponential growth and division in exponential decay.

Remember to be consistent with your units of time when using this formula. Also, note that this formula assumes a continuous growth or decay process, so it may not be suitable for all scenarios.

Example usage:

Let’s say you have an investment that has an initial value of $10,000 and grows at a rate of 5% per year. You want to find the value of the investment after 7 years.

Using the formula A(t) = P(1 + r)^t, we have:
A(7) = 10000(1 + 0.05)^7

Calculating that, we get:
A(7) = 10000(1.05)^7 ≈ 13,383.96

Therefore, the value of the investment after 7 years would be approximately $13,383.96.

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