Problem 7 - Algebra and Trigonometry

Question

Use mathematical induction to prove that the formula is true for all natural numbers n. $$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}$$

Step-by-Step Solution

Answer

The formula is proven for all natural numbers \(n\) by induction.

Step 1: Base Case Check

First, check the base case when \(n = 1\). Substitute \(n = 1\) into the formula: \[1^3 = \frac{1^2 (1+1)^2}{4} \] Evaluating both sides gives: \[1 = \frac{1 \times 4}{4} = 1\] The base case holds true.

Step 2: Inductive Hypothesis

Assume the formula is true for some arbitrary positive integer \(k\), i.e., \[1^3 + 2^3 + \cdots + k^3 = \frac{k^2 (k+1)^2}{4} \] This assumption is known as the inductive hypothesis.

Step 3: Inductive Step

To prove for \(n = k + 1\), consider \[1^3 + 2^3 + \cdots + k^3 + (k+1)^3 = \frac{k^2 (k+1)^2}{4} + (k+1)^3 \] Simplifying the right hand side gives: \[= \frac{k^2 (k+1)^2 + 4(k+1)^3}{4} \] Factor \((k+1)^2\) out: \[= \frac{(k+1)^2 (k^2 + 4(k+1))}{4} \] Simplifying inside the parentheses yields: \[= \frac{(k+1)^2 (k^2 + 4k + 4)}{4} \] Thus it can be rewritten as: \[= \frac{(k+1)^2 ((k+1)^2)}{4} \] Therefore, \[1^3 + 2^3 + \cdots + (k+1)^3 = \frac{(k+1)^2((k+2)^2)}{4} \] This matches the formula for \(n = k + 1\), completing the inductive step.

Step 4: Conclusion

Since both the base case and the inductive step have been proven, by mathematical induction, the formula \[1^3 + 2^3 + \cdots + n^3 = \frac{n^2 (n+1)^2}{4}\] is true for all natural numbers \(n\).

Key Concepts

Inductive Hypothesis

The inductive hypothesis is a pivotal part of proving statements using mathematical induction. To understand it better, think of it as making an assumption temporarily. In our given problem, we assume that the statement is true for some arbitrary positive integer \( k \).

This involves believing that the initial expression for \( k \) cubes:- \(1^3 + 2^3 + \cdots + k^3 = \frac{k^2(k+1)^2}{4}\) is valid. While you might think assuming something without evidence is unusual, in mathematics, this is a strategic move that bridges understanding from one integer to the next.

Essentially, the inductive hypothesis sets a foundation upon which we can build and prove the statement for the subsequent step, ensuring the hypothesis forms the core of our logical approach.

Base Case

The base case serves as the starting point for proving any statement by mathematical induction. It's like the first link in a chain, without which the entire linkage might collapse. For this problem, our base case is when \( n = 1 \).

We substitute \( n = 1 \) into our formula:- \(1^3 = \frac{1^2(1+1)^2}{4}\)

When we simplify both sides, - Everything balances as \( 1 = 1 \).

Confirming this initial case is crucial. This validation step assures us that the formula works at least for the first number in our series. The base case essentially supports the possibility to then proceed to the inductive step, ensuring continuity in the proof.

Inductive Step

The inductive step can be viewed as the crux of the induction process. This is where the magic happens, taking the assumption from the inductive hypothesis and using it to prove that if it holds for \( k \), it also holds for \( k + 1 \).

To make this transition in our problem, consider the expression involving \( k+1 \) cubes:

• \(1^3 + 2^3 + \cdots + k^3 + (k+1)^3 = \frac{k^2(k+1)^2}{4} + (k+1)^3\)
• Simplification allows us to factor \((k+1)^2\) and rearrange the terms:
  \(\frac{(k+1)^2(k^2+4k+4)}{4}\)
  and further
  \(\frac{(k+1)^2((k+1)^2)}{4}\)

This final step confirms our assumption extends to the next number in the sequence, for \( n = k + 1 \). This closure, that proves the formula is true for all \( n \), seals our mathematical induction, connecting the logical dots from base case to inductive step.

Algebraic Formulas

Algebraic formulas are fundamental tools in mathematical reasoning, especially in proofs such as induction. They provide the structure and rules that guide simplifications and transformations needed within our proof steps.

In the context of this exercise, algebraic manipulation is critical. We utilized operations to:

• Simplify complex polynomial expressions such as \( (k^2(k+1)^2) + 4(k+1)^3 \)
• Factor expressions like \( (k+1)^2 \) to ease computation and verify correctness

Being comfortable with such algebraic transformations not only helps in breaking down complex proofs but also strengthens the overall understanding of mathematical properties and their applications. Therefore, mastering these manipulations is key to tackling induction problems efficiently. This understanding supports every step from assuming the inductive hypothesis to verifying the inductive step.