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Chapter 13: Problem 20

Show that 3 is a factor of \(n^{3}+2 n\) for all natural numbers \(n\)

Short Answer

Expert verified
\(n^3 + 2n\) is divisible by 3 for all natural numbers \(n\) through mathematical induction.

Step by step solution

01

Understanding the Statement

We need to prove that the expression \(n^3 + 2n\) is divisible by 3 for any natural number \(n\). In other words, \(n^3 + 2n\) divided by 3 should leave no remainder.
02

Using Mathematical Induction (Base Case)

Let's start by verifying the base case with \(n = 1\). Substituting, we have: \(1^3 + 2(1) = 1 + 2 = 3\). Since 3 is divisible by 3, the base case is true.
03

Using Mathematical Induction (Inductive Step)

Assume that \(n^3 + 2n\) is divisible by 3 for some natural number \(n=k\). This means \(k^3 + 2k \equiv 0 \pmod{3}\). We need to show the same is true for \(n = k + 1\).
04

Prove Inductive Step

Consider \((k+1)^3 + 2(k+1)\). Expanding \((k+1)^3\), we get \(k^3 + 3k^2 + 3k + 1\). Therefore,\[(k+1)^3 + 2(k+1) = k^3 + 3k^2 + 3k + 1 + 2k + 2 = k^3 + 3k^2 + 5k + 3.\]
05

Simplifying and Verifying Divisibility

Notice that \(k^3 + 3k^2 + 5k + 3 = (k^3 + 2k) + 3(k^2 + k + 1)\). By the inductive hypothesis, \(k^3 + 2k\) is divisible by 3, and clearly, \(3(k^2 + k + 1)\) is divisible by 3 as it has a factor of 3. Therefore, \((k+1)^3 + 2(k+1)\) is also divisible by 3.
06

Conclusion

Since both the base case and the inductive step are verified, by the principle of mathematical induction, \(n^3 + 2n\) is divisible by 3 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.

Divisibility
Divisibility is an essential concept in mathematics, especially when dealing with algebraic expressions and proofs. When we say a number is divisible by another, it means dividing them results in a whole number with no remainder. For example, since 6 divided by 3 equals 2, 6 is divisible by 3.
In algebraic expressions, establishing divisibility helps us understand relationships between numbers. In our problem, we're asked to show that the expression \(n^3 + 2n\) is divisible by 3 for any natural number \(n\). This means if you divide \(n^3 + 2n\) by 3, you won't have a leftover 鈥� a crucial aspect of proving this with mathematical induction.
Proving divisibility often involves restructuring the expression or using specific mathematical techniques to confirm that after certain transformations, you are left with a factor (in our case, 3) of the expression.
Natural Numbers
Natural numbers are the basic building blocks of mathematics, starting from 1 and moving upwards (1, 2, 3, 4, ...). They are often used to count objects because they fit the concept of counting without fractions or decimals. Natural numbers do not include zero in this context, especially when used in mathematical proofs where dividing by a number remains valid.
In our problem, natural numbers are used to explore the expression \(n^3 + 2n\) for every possible natural number value. With concepts like mathematical induction, we can test a base case and a generic case \(n = k\), before proving the next number \(k + 1\). In this way, we demonstrate that our statement holds true for all natural numbers.
Algebraic Expressions
Algebraic expressions combine numbers, variables, and operators to describe mathematical relationships. For instance, the expression \(n^3 + 2n\) involves the variable \(n\), which represents any natural number.
These expressions can be manipulated to uncover underlying patterns or properties, such as divisibility. Simplifying expressions or expanding them, as in our problem, is a common strategy, especially when using induction for proofs.
Expanding \((k + 1)^3\) involves breaking it down into terms that can be recombined or simplified. By doing this, we're able to observe factors, like 3, emerge, which helps confirm divisibility. Learning to manage algebraic expressions skillfully is crucial for solving complex problems and forming mathematical arguments.

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