Bernoulli Inequality Calculator

What is the Bernoulli Inequality?

The Bernoulli Inequality is like a special rule in math. It tells us that when we add 1 to a number and then raise it to a power, the result is always bigger than (or equal to) just adding that power times the number to 1. It's a bit like saying that compound interest grows faster than simple interest!

How to Calculate the Bernoulli Inequality

To check if the Bernoulli Inequality is true, we compare two sides of an equation. On one side, we add 1 to a number and raise it to a power. On the other side, we multiply that number by the power and add 1. If the first side is bigger or equal, the inequality holds true!

Formula

The Bernoulli Inequality states that for any real number \(x > -1\) and any integer \(r \geq 0\):

\[ (1 + x)^r \geq 1 + rx \]

Where:

• \(x\) is any real number greater than -1
• \(r\) is any non-negative integer

Calculation Steps

1. Choose a value for \(x\) (remember, it should be greater than -1)
2. Choose a non-negative integer for \(r\)
3. Calculate the left side: \((1 + x)^r\)
4. Calculate the right side: \(1 + rx\)
5. Compare the two sides. If the left side is greater than or equal to the right side, the inequality holds!

Example

Let's try with \(x = 0.5\) and \(r = 2\)

1. Left side: \((1 + 0.5)^2 = 1.5^2 = 2.25\)
2. Right side: \(1 + (2 \times 0.5) = 1 + 1 = 2\)
3. We see that 2.25 > 2, so the inequality holds!

Visual Representation