Bernoulli Inequality Mathematical Induction Calculator

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 Bernoulli Inequality Mathematical Induction =

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The Bernoulli inequality is a useful tool in mathematical induction, allowing you to prove inequalities for all natural numbers. Our calculator makes it easy to apply the Bernoulli inequality to your own mathematical proofs. Here's how to use it:

Instructions:

1. Enter the base case value
a0
2. Enter the inequality statement you want to prove in terms of
an

Our calculator will apply the Bernoulli inequality to each step of the induction and display the resulting inequality for each step. You can use this proof as a reference for your own mathematical proofs.

Formula:

The Bernoulli inequality states that for any real number

x ≥ -1
and any natural number
n ≥ 0
, the inequality
(1+x)^n ≥ 1+nx
holds.

Example

Let's say we want to prove that the sequence defined by an = (1 + 1/n)n satisfies the inequality an < e for all n ≥ 1. To use the calculator, we would follow these steps:

Step 1: Enter "n=5" for the number of terms in the sequence.

Step 2: Enter "a1=(1 + 1/1)^1" for the initial value of the sequence.

Step 3: Enter "x^n < e" for the inequality we want to prove, since we want to prove an < e for all n ≥ 1.

The calculator will then use mathematical induction to prove that the inequality an < e holds for all terms of the sequence.

Formula

The formula for the Bernoulli Inequality is:

(1 + x)n ≥ 1 + nx for all x ≥ -1 and all n ≥ 0.

Conclusion

The Bernoulli Inequality Mathematical Induction Calculator is a useful tool for proving that a sequence of numbers satisfies a particular inequality using mathematical induction. By following the steps outlined in this tutorial, you can easily use the calculator to prove the desired inequality for any sequence.

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