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Chapter 2: Problem 2

Prove that $$ 1^{3}+2^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4} $$ for \(n \in \mathbb{N}\).

Short Answer

Expert verified
Question: Prove that for any natural number n, the summation of cubes from 1 to n is equal to the square of the arithmetic mean of 1 to n. In other words, prove that $$ 1^3+2^3+\cdots+n^3=\frac{n^2(n+1)^2}{4}. $$ Answer: The statement has been proven true for all natural numbers n using the induction method.

Step by step solution

01

Base case (n=1)

For n=1, we have only one term in the summation: \(1^3\). The right side of the equation is \(\frac{1^2(1+1)^2}{4}\). Let's compare both sides: $$ 1^3 = 1=\frac{1^2(1+1)^2}{4}=\frac{4}{4} $$ Both sides are equal, so the base case is true.
02

Inductive Hypothesis

Now we assume that the statement is true for some arbitrary natural number \(k\). That is, we assume that: $$ 1^3+2^3+\cdots+k^3=\frac{k^2(k+1)^2}{4} $$
03

Prove for n=k+1

We want to prove the statement for n=k+1. That is, we want to show that: $$ 1^3+2^3+\cdots+k^3+(k+1)^3=\frac{(k+1)^2(k+2)^2}{4} $$ Based on our inductive hypothesis, we can substitute the statement for n=k into this equation: $$ \frac{k^2(k+1)^2}{4}+(k+1)^3=\frac{(k+1)^2(k+2)^2}{4} $$ Now, let's manipulate the equation and see if both sides are equal.
04

Simplify the equation

Take out a common factor of \((k+1)^2\) on the left side and simplify the equation: $$ \frac{k^2(k+1)^2}{4}+(k+1)^3 = \frac{(k+1)^2}{4}(k^2+4(k+1)) $$ Now we need to show that the right side of the equation is equal to \(\frac{(k+1)^2(k+2)^2}{4}\): $$ \frac{(k+1)^2}{4}(k^2+4(k+1)) = \frac{(k+1)^2(k^2+4k+4)}{4} $$ We can see that both sides are equal.
05

Conclusion

Since the base case is true, and we have proven the induction step, the statement: $$ 1^3+2^3+\cdots+n^3=\frac{n^2(n+1)^2}{4} $$ is true for all natural numbers n.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proof Techniques
Mathematical induction is a crucial proof technique used to show that a property holds for all natural numbers. It is particularly useful for proving the validity of sequences and series, like the sum of cubes formula presented in the exercise. Induction involves two main steps: the base case and the inductive step.

For the base case, we verify that the property holds for the initial value, often n=1. Once the base case is confirmed, we assume that the property holds true for some arbitrary natural number k; this is called the inductive hypothesis. The inductive step involves proving that if the property holds for k, it must hold for k+1 as well. By successfully showing this, we can state that the property holds for all natural numbers succeeding the base case, thus completing the induction process.

Inductive proofs can sometimes be intricate and require good algebraic manipulation skills. This makes educational content that simplifies understanding of each step valuable for students grappling with mathematical proofs.
Algebraic Proofs
Algebraic proofs are logically structured arguments that use algebraic techniques to demonstrate the truth of a given assertion. In the context of the exercise, algebra is used to manipulate formulas and simplify expressions to prove that for some natural number k, the expression involving k is logically extended to k+1.

In the given solution, algebraic operations are essential to show the equality of two sides in the inductive step. For example, factoring, expanding expressions, and simplifying are algebraic techniques used in the transition from step 3 to step 4 of the proof. It's important for students to understand not just the operations but the rationale behind each step. Clear explanations and breaking down complex operations into smaller, easily digestible parts can enhance comprehension.

Incorporating practice problems that progressively increase in difficulty can nurture students' algebraic reasoning and improve their ability to craft sound algebraic proofs.
Mathematical Formulas
The exercise demonstrates a classic example of how mathematical formulas represent relationships among quantities. Formulas are vital tools in solving problems and proving various mathematical statements. Knowing how to derive and manipulate these formulas is a key skill in advanced mathematics.

The formula in this exercise, \(1^3+2^3+\cdots+n^3=\frac{n^2(n+1)^2}{4}\), illustrates the sum of cubes of the first n natural numbers. Understanding how to derive such a formula requires grasping both the pattern it represents and the mechanism of proof that establishes its validity. Mathematical formulas are not just memorized; they are proven and understood using logical structure and mathematical laws.

To aid understanding, providing interactive examples or visual aids to represent the progression from \(1^3\) to \(n^3\) can help students visualize the sum and the resulting formula. Exercises that guide students in exploring and discovering the beauty of these formulas nurture their intuition and appreciation for the coherence and elegance of mathematics.

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Most popular questions from this chapter

Power Sets. \(\quad\) Let \(X\) be a set. Define the power set of \(X,\) denoted \(\mathcal{P}(X),\) to be the set of all subsets of \(X\). For example, $$ \mathcal{P}(\\{a, b\\})=\\{\emptyset,\\{a\\},\\{b\\},\\{a, b\\}\\} $$ For every positive integer \(n\), show that a set with exactly \(n\) elements has a power set with exactly \(2^{n}\) elements.

Let \(a, b, c \in \mathbb{Z}\). Prove that if \(\operatorname{gcd}(a, b)=1\) and \(a \mid b c\), then \(a \mid c\).

Let \(a\) and \(b\) be integers such that \(\operatorname{gcd}(a, b)=1\). Let \(r\) and \(s\) be integers such that \(a r+b s=1 .\) Prove that $$ \operatorname{gcd}(a, s)=\operatorname{gcd}(r, b)=\operatorname{gcd}(r, s)=1 $$

Prove that \(n !>2^{n}\) for \(n \geq 4\).

If \(d=\operatorname{gcd}(a, b)\) and \(m=\operatorname{lcm}(a, b),\) prove that \(d m=|a b| .\)
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