Problem 20 - Fundamentals of Mathematics

Question

The sum of five consecutive integers is zero. What are they?

Step-by-Step Solution

Answer

The integers are -2, -1, 0, 1, and 2.

Step 1: Define the Unknowns

Let the five consecutive integers be represented as \( x, x+1, x+2, x+3, \) and \( x+4 \), where \( x \) is the starting integer.

Step 2: Set Up the Equation

Since the sum of these five consecutive integers is zero, we can write the equation: \( x + (x+1) + (x+2) + (x+3) + (x+4) = 0 \).

Step 3: Simplify the Equation

Combine like terms in the equation: \( 5x + 10 = 0 \).

Step 4: Solve for x

Subtract 10 from both sides to get \( 5x = -10 \). Then, divide both sides by 5 to solve for \( x \): \( x = -2 \).

Step 5: Identify the Integers

Substitute \( x = -2 \) back into the expressions for the five consecutive integers. They are \( -2, -1, 0, 1, \) and \( 2 \).

Key Concepts

Sum of Integers

When we talk about the **sum of integers**, we mean adding up a series of whole numbers. In math problems, you often see questions about the sum of consecutive integers, like the five numbers in our exercise. Consecutive integers are numbers that follow each other in order, like 1, 2, 3, 4, and 5. In our case, these integers played a special role: their sum was zero. That means the integers include both negative and positive numbers, which cancel each other out to give a total of zero.

• Consecutive integers have a pattern, where each number is one more than the previous one.
• The sum of a mix of positive and negative consecutive integers equal to zero suggests symmetry around the center of the sequence.

Understanding how their individual values work collectively can help when solving related math problems, especially when working with equations.

Equations with Integers

**Equations with integers** are a fundamental part of math, serving as the backbone for solving problems involving unknowns. In this scenario, we constructed an equation to find the integers. Each unknown is represented by a variable (often denoted by letters like \( x \)). For five consecutive integers, we used variables: \( x, x+1, x+2, x+3, \) and \( x+4 \).

• The equation formed from these variables helps us encapsulate all the information about the sum of these integers.
• By writing it as \( x + (x+1) + (x+2) + (x+3) + (x+4) = 0 \), we link each integer's position to the initial unknown \( x \).

Setting up equations with integers like this is a powerful tool to break down complex problems into manageable parts.

Solving Equations

**Solving equations** is about finding the value of unknowns that satisfy the equation. We started with the long equation from our problem and simplified it by combining like terms. This step reduces clutter and focuses on key parts.

• Combining terms like \( x \)s and numbers separately can make your life easier. We summed them into a single equation: \( 5x + 10 = 0 \).
• Next, isolate the variable \( x \) by first subtracting the constant term from both sides, which gives us \( 5x = -10 \).
• Finally, divide by the coefficient of \( x \), which is 5, to solve for \( x = -2 \).

Breaking down each part in solving lets you focus on finding a solution step by step without mistakes. It leaves you with clear numbers you can work with.

Integer Properties

Knowing about **integer properties** gives us insights into how numbers behave. Integers include all whole numbers, both positive and negative, including zero. Understanding these properties is key in many math problems.

• Integers have a natural order: from lower to higher (including negative numbers).
• Adding or subtracting integers follows straightforward rules – with opposites cancelling out (like \(-2 + 2 = 0\), where they balance each other).
• This balancing act is exactly what happens for the five consecutive integers, which sum to zero.

In our exercise, knowing that integers' properties help balance positive and negative sums can clarify why the sequence \(-2, -1, 0, 1,\) and \(2\) works.