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Mathematical Induction Calculator

Enter x value and power in mathematical induction calculator to prove the Bernoulli's inequality for any function.

Bernoulli's Inequality

Mathematical Induction Calculator

Mathematical induction calculator is an online tool that proves the Bernoulli's inequality by taking x value and power as input. This induction proof calculator proves the inequality of Bernoulli’s equation by showing you the step by step calculation.

What is mathematical induction?

Mathematical induction is a mathematical proof technique. It is a technique for proving results or establishing statements for natural numbers.

The procedure requires two steps to prove a statement.

Step 1: It proves that a statement is true for the initial value. It is a base step.

Step 2: It proves that if the statement is true for the nth iteration (or number n), then it is also true for (n+1) th iteration (or number n+1). It is an inductive step.

Bernoulli's inequality – What is it?

Bernoulli's inequality is named after Jacob Bernoulli. In mathematics, it is an inequality that approximates exponentiations of 1 + x. It is mostly used in real analysis.

Formula

(1 + x)r ≥ 1 + rx
Where,
x refers to the real numbers and x ≥ -1,

r refers to the real number and r ≠ 0.

How to prove Bernoulli’s inequality?

Even though mathematical induction solver can prove any Bernoulli’s inequality, you should also go through the step by step method. We will explain it with an example below.

Example

Prove: 1 + 3 + 5 + ... + (2n−1) = n2

Step 1: Let’s check if it is true for n=1.

1 = 12 ---------- True 

Step 2: Let’s suppose it is true for n=k.

1 + 3 + 5 + ... + (2k−1) = k2 ----------> True
Note that, it is only an assumption.

Step 3: Let’s prove it is true for "k+1".

1 + 3 + 5 + ... + (2k−1) + (2(k+1) −1) = (k+1)2

As we already know that

1 + 3 + 5 + ... + (2k−1) = k2

So, we can do a replacement for the whole expression except the last term in it.

k2 + (2(k+1) −1) = (k+1)2

Step 4: Let’s expand all terms in the expression.

k2 + 2k + 2 − 1 = k2 + 2k+1

After simplifying, we get…

k2 + 2k + 1 = k2 + 2k + 1

As you can see, both sides are same. It means it is true.

1 + 3 + 5 + ... + (2(k+1) −1) = (k+1)2 ----------> True

You can use math induction calculator above to save yourself from these lengthy calculations.