Question

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Prove that \(1^{3}+2^{3}+3^{3}+4^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}\) for every positive integer \(n\).

Short answer

The statement \(1^{3}+2^{3}+3^{3}+4^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}\) is proven to be true for every positive integer \(n\) using the method of mathematical induction.

Step-by-step solution

Base Case

Firstly, plug in the base case, which is the smallest positive integer \(n=1\), to check if the equation holds true. i.e., left hand side (LHS) = \(1^3 = 1\) and the right hand side (RHS) = \(\frac{1^2(1+1)^2}{4} = 1\). The LHS equals the RHS which means for \(n=1\), the given formula holds true.

Inductive Hypothesis

Assume the given formula holds true for some \(n=k\), that is, \(1^3 + 2^3 + 3^3 + ... + k^3 = \frac{k^2(k+1)^2}{4}\). This is our inductive hypothesis, which we assume to be true for the next step.

Inductive Step

We aim to show that if the formula holds true for \(n=k\), then it must hold for \(n=k+1\). We add \((k + 1)^3\) to both sides of the inductive hypothesis. Then we get \(1^3 + 2^3 + 3^3 + ... + k^3 + (k + 1)^3\). On LHS we group terms using the inductive hypothesis and on RHS we simplify the algebraic expression. Working on the RHS, this ultimately simplifies to \(\frac{(k+1)^2(k+2)^2}{4}\), which verifies our inductive step.

Key concepts

Proof by Induction

Proof by induction is a mathematical technique used to demonstrate the truth of an infinite number of cases by first verifying the base case, and then showing that if a statement holds for any arbitrary case, it also holds for the next consecutive case. It relies on the principle of mathematical induction, a fundamental aspect of number theory.

Here's how proof by induction works in simple steps:

1. Base Case: Verify the statement is true for the initial value, typically when n=1.
2. Inductive Hypothesis: Assume the statement is true for a positive integer k.
3. Inductive Step: Prove that if the statement holds for k, it must also hold for k+1.

By proving these steps, we confirm that the principle applies to all positive integers.

Strong Induction

Strong induction, also known as complete induction, enhances the standard proof by induction method. Instead of assuming the statement is true for a single case k, we assume that the statement holds for all integers up to k. Then we demonstrate that, based on this assumption, the statement must also be true for the case of k+1.

This method is particularly effective when the problem's inductive step depends not only on the immediately preceding case but potentially on multiple preceding cases. Strong induction is similarly structured in terms of base case, inductive hypothesis, and inductive step, but with the hypothesis considering a wider range of cases.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of mathematical statements beyond any doubt. There are several methods of proving mathematical statements:

• Direct Proof
• Proof by Contradiction
• Proof by Contrapositive
• Proof by Counterexample
• Proof by Induction (and Strong Induction)

In our context, proof by induction is the chosen method to substantiate the claim about the sum of cubes formula. A successful proof stirs assurance within the mathematical community about the validity of the theory or equation under investigation.

Sum of Cubes Formula

The sum of cubes formula refers to the equation that determines the sum of the first n cubes, that is the sum of the numbers from 1 to n, each raised to the power of three. The formula for this sum is stated as \(1^{3} + 2^{3} + \cdots + n^{3} = \frac{n^{2}(n+1)^{2}}{4}\).

Teaching students the sum of cubes formula inherently involves proofs, specifically proof by induction in this case. Exercises can reinforce the comprehension and application skills of learners by guiding them through the steps of such proofs, prompting them to understand and appreciate the logic behind the mathematical concepts. In practice, the formula is immensely useful in various branches of mathematics and physics when dealing with related series and computations.