91Ó°ÊÓ

Understanding divisibility is key in mathematical proofs. When we say a number is divisible by another, it means that dividing these numbers yields a whole number without any remainder. In our problem, we need to prove that \(n^3 + 5n\) is divisible by 6, meaning that using any natural number \(n\) the resulting expression should leave no remainder when divided by 6.

To show divisibility, you can check whether the expression is evenly divisible by both 2 and 3, which are factors of 6. For instance, if both \(n^3\) and \(5n\) result in even numbers, their sum is divisible by 2. Similarly, if \(n^3 + 5n\) is also divisible by 3, then it can be concluded that the expression is divisible by 6.

• Consider even and odd values of \(n\). Try calculating \(n^3 + 5n\) for small numbers to see patterns.
• Use algebra to regroup terms to show how both parts contribute to divisibility by 6.

Dividing expressions into components can help prove their overall divisibility.

Natural numbers are the foundational concept used in this problem. By definition, natural numbers are positive integers starting from 1. They form the base upon which mathematical induction operates. Induction is a crucial technique in this proof, as we claim that a statement is true for all natural numbers.

When starting with the smallest natural number, \(n = 1\), you can verify initial conditions. This is known as the base case, and it's crucial because it sets the foundation for further steps in mathematical induction.

• The sequence of natural numbers increases incrementally (1, 2, 3,…), simplifying broader applications in proofs.
• Natural numbers ensure we never deal with fractions or negative numbers, sticking strictly to integers.

This guarantee simplifies initial testing and the application of the inductive method.

Algebraic proofs involve using algebraic manipulation to demonstrate the truth of a statement. In this exercise, algebraic techniques are harnessed to support the induction proof for divisibility.

The key steps in crafting an algebraic proof include setting up a base case, forming an induction hypothesis, and performing an inductive step.

• Base Case: Check the smallest instance where \(n=1\). This verification shows that the initial statement holds true.
• Induction Hypothesis: Assume the statement is true for \(n=k\), such as \(k^3 + 5k\) is divisible by 6.
• Inductive Step: Prove for \(n=k+1\). Simplify expressions using assumptions and algebraic manipulation to generalize the proof.

This approach unifies logical reasoning and algebra to construct valid conclusions.