Problem 8 - Precalculus Mathematics for Calculus

Question

Express the given quantity in terms of the indicated variable. The sum of three consecutive integers; \(n=\) middle integer of the three.

Step-by-Step Solution

Answer

The sum is \(3n\).

Step 1: Identify Consecutive Integers

Consecutive integers are numbers that follow each other without any gap. If the middle integer is represented as \(n\), the three consecutive integers can be written as \(n-1\), \(n\), and \(n+1\).

Step 2: Write Expression for the Sum

The sum of the three consecutive integers \((n-1), n, (n+1)\) can be expressed as follows: \((n-1) + n + (n+1)\).

Step 3: Simplify the Expression

Combine the terms of the sum: \((n-1) + n + (n+1) = n - 1 + n + n + 1\). Simplify it to \(3n\).

Key Concepts

Sum of Integers

When dealing with consecutive integers, the sum plays a vital role in combining and understanding the relationship among them. Consecutive integers are those numbers that follow each other without gaps, like 3, 4, and 5. The sum of such integers is an interesting exercise that helps in problem-solving and understanding number patterns.
In this exercise, we are solving for the sum of three consecutive integers where the middle one is given as a variable, \(n\). The numbers can be defined as \(n-1\), \(n\), and \(n+1\). Their sum is represented by the expression:

• \((n-1) + n + (n+1)\)

This illustrates a simple principle: whether you add a starting point, a middle, or an endpoint, the cumulative total involves all variations.
Adding these integers together simplifies to \(3n\), which shows how easily consecutive numbers can be represented in a compact mathematical form.

Variable Representation

Variables stand as placeholders that can represent a wide range of values, and they are pivotal in forming equations and expressions. When you have a variable, such as \(n\), you are creating a flexible component within math that allows for dynamic problem-solving.
In this exercise, using \(n\) to denote the middle integer makes the task of handling consecutive numbers systematic and neat. This way, rather than writing out each integer individually, you capture them through their relationship to the variable:

• \(n-1\): Represents the integer just before \(n\).
• \(n\): Stands as the middle integer.
• \(n+1\): Indicates the integer just after \(n\).

This representation not only simplifies calculations but also helps in visualizing the numbers in a coherent structure.
Through variable representation, we gain a better understanding of numerical arrangements and can manipulate these arrangements to derive desired outcomes such as sums.

Algebraic Expression

Algebraic expressions form the bedrock of algebra, allowing us to translate written problems into mathematical language. In this particular case, the algebraic expression emerges to articulate the sum of three consecutive integers.
We start with a clear representation of the integers using the variable \(n\), which then converts into an expression. The full expression for their sum is:

• \((n-1) + n + (n+1)\)

By simplifying this expression, we combine like terms. This involves adding the coefficients of \(n\) and constants separately, leading to the final simplified expression:

• \(3n\)

The beauty of algebraic expressions is their ability to succinctly embody the essence of a problem.
They show how the values interrelate and provide a powerful tool for solving complex problems efficiently using simple algebraic manipulations.