Problem 15 - Precalculus Mathematics for Calculus

Question

Show that \(n^{2}+n\) is divisible by 2 for all natural numbers \(n\)

Step-by-Step Solution

Answer

\(n^2 + n\) is divisible by 2 because it is the product of two consecutive integers, one of which is always even.

Step 1: Express the Equation

We want to show that \( n^2 + n \) is divisible by 2 for all natural numbers \( n \). To do this, rewrite the expression as \( n(n + 1) \).

Step 2: Identify the Nature of Successive Integers

Notice that \( n \) and \( n+1 \) are two consecutive integers. For any two consecutive integers, one of them must be even.

Step 3: Analyze the Divisibility

Since either \( n \) or \( n + 1 \) must be even, their product \( n(n+1) \), which is the same as \( n^2+n \), is necessarily even. This implies it is divisible by 2.

Key Concepts

Divisibility

Divisibility is a useful concept in mathematics, especially in number theory. It describes the ability of one number to be divided by another without a remainder. In simple terms, if a number \(a\) can be divided by another number \(b\), such that the result is an integer, \(a\) is said to be divisible by \(b\). For example, 10 is divisible by 5 because \(10 \div 5 = 2\), which is an integer.

Checking divisibility often involves identifying patterns or characteristics in numbers. For instance, any even number is divisible by 2, because it can be divided by 2 with no remainder. Understanding these patterns simplifies the process of determining divisibility, as seen in exercises like proving \(n^2 + n\) is divisible by 2.

• Key point: divisibility by 2 means a number is even.
• If a number is even, it can be expressed as \(2k\), where \(k\) is an integer.

Even and Odd Numbers

Numbers can be classified as either even or odd based on their divisibility by 2. An even number is any integer that can be divided by 2 with no remainder. Odd numbers, on the other hand, have a remainder of 1 when divided by 2. For example, 4 is even because \(4 = 2 \times 2\), whereas 5 is odd because \(5 = 2 \times 2 + 1\).

In mathematics, understanding whether numbers are even or odd can help establish important properties, like invariants or dissect patterns in sequences. A sequence of numbers formed by alternating between even and odd characteristics tends to reveal interesting mathematical truths. For instance, in the expression \(n^2 + n\), when \(n\) is even, \(n^2\) is even, and \(n\) is even, making \(n^2 + n\) even. When \(n\) is odd, \(n + 1\) is even, thus \(n(n+1)\) is still even.

• Consecutive numbers include one even and one odd number.
• The product of an even and any integer is always even.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of mathematical statements. A proof must show that a statement is true for all possible cases, using a step-by-step approach detailed by mathematical reasoning.

There are various types of proofs, like direct proofs, contradiction, and induction. For the problem of showing \(n^2 + n\) is divisible by 2, a direct proof works effectively. By expressing the polynomial as \(n(n + 1)\), it’s evident that successive numbers encompass both even and odd values. This turns the product into an even number because at least one part of the multiplication is guaranteed to be even.

Successful proofs not only conclude with logical certainty but also simplify intricate problems into understandable solutions.

• Direct proofs involve assuming conditions and deriving conclusions logically.
• They clarify conditions and transform statements into evident calculations.