Problem 10 - A Gentle Introduction to the Art of Mathematics

Question

Give inductive proofs of the following $$ \forall n \geq 3 \in \mathbb{N}, 3 n^{2}+3 n+1<2 n^{3} $$

Step-by-Step Solution

Answer

By mathematical induction, the inequality holds for all natural numbers n ≥ 3.

Step 1 - Base Case

Check the base case where n=3. Substitute 3 into the inequality and verify if it holds. Substitute: \[3(3)^{2} + 3(3) + 1 < 2(3)^{3}\] Calculate: \[27 + 9 + 1 < 54 \] Observe: \[37 < 54 \] Thus, the base case holds true.

Step 2 - Inductive Hypothesis

Assume that the inequality holds for some integer k such that k ≥ 3. That is, assume: \[ 3k^2 + 3k + 1 < 2k^3 \] This is the inductive hypothesis.

Step 3 - Prove for k+1

Next, prove that if the inequality holds for k, it also holds for k + 1. Start by expressing the inequality for k + 1: \[ 3(k+1)^2 + 3(k+1) + 1 < 2(k+1)^3 \] Expand and simplify the left side: \[ 3(k^2 + 2k + 1) + 3k + 3 + 1 \] Combine like terms: \[ 3k^2 + 6k + 3 + 3k + 3 + 1 \] \[ 3k^2 + 9k + 7 \] For the right side, expand: \[ 2(k^3 + 3k^2 + 3k + 1) \] \[ 2k^3 + 6k^2 + 6k + 2 \] We need to show: \[ 3k^2 + 9k + 7 < 2k^3 + 6k^2 + 6k + 2 \] This simplifies to: \[ 0 < 2k^3 + 3k^2 - 3k - 5 \] Since both sides are clearly positive for all k ≥ 3, the inequality holds.

Step 4 - Conclusion

By mathematical induction, the given inequality holds for all natural numbers n ≥ 3.

Key Concepts

Mathematical Induction

Mathematical induction is a powerful technique for proving statements about all natural numbers. Think of it like dominoes. If you push over one domino, and if each domino pushes the next one, all the dominoes will fall. In math, to prove a statement using induction, we show that if one case holds, then the next one does too. This involves:
- Proving the base case (the first domino falls)
- Proving the inductive step (if one domino falls, the next one falls)
This ensures that the statement is true for all numbers following the base case.

Base Case

The base case is the first step in a proof by induction. It means proving that the statement is true for the smallest value you are interested in. In our exercise, the base case is for n=3. Here's how it works:

• Substitute n=3 into the inequality: \[3(3)^{2} + 3(3) + 1 < 2(3)^{3}\]


• Simplify it: \[27 + 9 + 1 < 54\]


• This gives 37 < 54, which is true.

Since the base case holds, we can proceed to the next step.

Inductive Hypothesis

The inductive hypothesis is the assumption we make to help push the 'dominoes' of our proof. We assume the statement is true for some arbitrary number k. In our case, we assume:
\[ 3k^2 + 3k + 1 < 2k^3 \]
This assumption forms the basis for proving that if it holds for k, it must also hold for k+1. Think of it like assuming one specific domino falls to show that it will push the next one over.

Inequalities in Mathematics

Inequalities are a fundamental concept in mathematics used to compare two values or expressions. In our inductive proof, we deal with the inequality \[ 3n^{2}+3n+1 < 2n^{3} \].
To prove this inequality, we:

• First, verify it for the base case (n=3).
• Then assume it holds true for k (inductive hypothesis).
• Finally, we prove it holds for k+1.

By handling inequalities carefully, we show a mathematical statement holds true across a range of values, providing a clear and rigorous proof. The key is to systematically break down and simplify expressions to make comparisons easier.