Proof of Inequality by Induction

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Start date Jul 2, 2013
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Induction Inequality Proof

In summary, "Proof of Inequality by Induction" is a mathematical method used to prove that an inequality is true for all values of a variable by using a specific set of steps and assumptions. It differs from regular induction in that it is used to prove statements about real numbers. The steps involved in this method include a base case, inductive hypothesis, inductive step, and conclusion. It is typically used when trying to prove a statement involving real numbers. However, it may have limitations, such as difficulty in finding a suitable inductive hypothesis and being limited to statements involving real numbers.

Jul 2, 2013

sbrajagopal2690

need help on this

Show by induction that n^3 <= 3^n for all natural numbers n.

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Jul 2, 2013

Opalg

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MHB

Re: Proof of Inequlity by Induction

sbrajagopal2690 said:

need help on this

Show by induction that n^3 <= 3^n for all natural numbers n.

Hint: If $n\geqslant 3$ then $(n+1)^3 \leqslant \bigl(n+\frac n3\bigr)^3$.

Related to Proof of Inequality by Induction

1. What is "Proof of Inequality by Induction"?

"Proof of Inequality by Induction" is a mathematical method used to prove that an inequality is true for all values of a variable by using a specific set of steps and assumptions.

2. How does "Proof of Inequality by Induction" differ from regular induction?

The main difference between "Proof of Inequality by Induction" and regular induction is that regular induction is used to prove statements about whole numbers (usually starting at 1), while "Proof of Inequality by Induction" is used to prove statements about real numbers.

3. What are the steps involved in a "Proof of Inequality by Induction"?

The steps involved in a "Proof of Inequality by Induction" are:

Base Case: Prove that the inequality is true for the first value of the variable.
Inductive Hypothesis: Assume that the inequality is true for a specific value of the variable.
Inductive Step: Use the inductive hypothesis to prove that the inequality is true for the next value of the variable.
Conclusion: Use the inductive step to show that the inequality is true for all subsequent values of the variable.

4. When is "Proof of Inequality by Induction" used?

"Proof of Inequality by Induction" is used when trying to prove a statement involving real numbers, such as inequalities, summations, or product formulas.

5. What are the potential limitations of using "Proof of Inequality by Induction"?

One potential limitation of using "Proof of Inequality by Induction" is that it can be difficult to find a suitable inductive hypothesis, and if the hypothesis is not strong enough, the proof may fail. Additionally, "Proof of Inequality by Induction" can only be used on statements that involve real numbers, so it may not be applicable in all mathematical scenarios.