Problem 7 - Algebra and Trigonometry

Question

\(7-18 \cdot\) Express the given quantity in terms of the indicated variable. The sum of three consecutive integers; \(\quad n=\) first integer of the three

Step-by-Step Solution

Answer

The expression is \(3n + 3\).

Step 1: Understanding Consecutive Integers

Consecutive integers are numbers that follow each other in order without any gaps. If we denote the first integer as \(n\), the next consecutive integers would be \(n+1\) and \(n+2\).

Step 2: Setting Up the Expression

To find the sum of three consecutive integers, we need to add them together. Given the integers: \(n\), \(n+1\), and \(n+2\), the sum would be expressed as follows: \[ n + (n+1) + (n+2) \]

Step 3: Simplifying the Expression

Combine the like terms in the expression from Step 2. This results in: \[ n + n + 1 + n + 2 = 3n + 3\]

Step 4: Final Expression

The simplified expression for the sum of the three consecutive integers, in terms of \(n\), is: \[ 3n + 3\]

Key Concepts

Sum of Integers

When we talk about the sum of integers, we mean adding together a series of whole numbers. In this exercise, we focus on consecutive integers, which are numbers that immediately follow one another, like 5, 6, and 7. Calculating the sum involves simply adding these numbers. For the specific problem of three consecutive integers, we start with the first integer, denoted as \(n\), and then add the next two consecutive numbers: \(n+1\) and \(n+2\).
This gives us the equation \(n + (n+1) + (n+2)\). Keeping track of the order of numbers and carefully adding them ensures we correctly calculate the total sum.

Variable Expression

A variable expression is a mathematical phrase that can include numbers, variables (like \(n\)), and operations (such as addition and multiplication). Variables represent numbers that can change or vary, hence the name. In problems involving expressions, like the sum of consecutive integers, using a variable helps us generalize the expression.
Here, we use \(n\) to represent the first integer. This representation allows us to write a formula that works for any starting number. The expression \(n + (n+1) + (n+2)\) shows how consecutive numbers build on the variable \(n\), offering a flexible way to calculate sums for any starting integer.
Understanding how variable expressions work gives us insights into writing formulas and solving problems more easily.

Simplifying Expressions

Simplifying expressions is about making a mathematical phrase shorter and easier to work with while keeping its value the same. This often means combining like terms, which are terms in an expression that have the same variable parts. For the expression \(n + (n+1) + (n+2)\), we have multiple \(n\) terms that can be grouped together.
We start by identifying these like terms: \(n, n, \) and another \(n\). Adding them together simplifies the expression to \(3n\). Similarly, we add the constant numbers \(1\) and \(2\) to get \(3\).
The entire expression of \(n + (n+1) + (n+2)\) then simplifies to \(3n + 3\). Simplification not only makes the expression cleaner but also helps in making calculations straightforward and fast.