Problem 2 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The sum of the integers is \(3n\).

Step 1: Understand the Problem

We are given three consecutive integers and need to express their sum in terms of a variable. The problem specifies that the middle integer is represented by \(n\). The task is to find an algebraic expression for the sum of these integers.

Step 2: Define the Consecutive Integers

For any set of three consecutive integers, if the middle one is \(n\), the previous integer will be \(n-1\) and the next integer will be \(n+1\).

Step 3: Express the Sum

We now express the sum of these integers as \((n-1) + n + (n+1)\).

Step 4: Simplify the Expression

By simplifying, combine like terms in the expression: \[(n-1) + n + (n+1) = 3n\]Here, \(-1\) and \(+1\) cancel each other.

Step 5: Final Result

The sum of the three consecutive integers is \(3n\), where \(n\) is the middle integer.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. In this context, we're using the expression to represent the sum of integers. A variable, such as \(n\), serves as a placeholder for unknown values. Here, \(n\) represents the middle integer in a sequence of three consecutive integers. This means when we talk about the algebraic expression for the sum of these integers, it reflects how the sum will change depending on the value of \(n\).
For instance, if you imagine \(n\) being 5, then the integers would be 4, 5, and 6. The algebraic expression for their sum helps us see the relationship between the quantities involved without having to plug in numbers for every case.

Simplifying Expressions

Simplifying an expression is the process of making it as straightforward as possible while retaining its original meaning. For our problem, the initial expression for the sum of three consecutive integers is \((n-1) + n + (n+1)\).

To simplify:

• First, observe the integers in the expression: \(n-1\), \(n\), and \(n+1\).
• Combine all the \(n\) terms: \(n\) appears three times, resulting in \(3n\).
• Next, simplify the constants: \(-1\) and \(+1\) cancel each other out.

Thus, the simplified form is \(3n\). This represents the sum in the most reduced manner, where \(n\) is the key variable.

Integer Sequences

Integer sequences involve a series of numbers with a common pattern or rule. Consecutive integers are a simple form of integer sequences where each number is one more than the previous one. In this case, the sequence is of three consecutive numbers.
The middle integer \(n\) specifies the position in this sequence. Once you understand \(n\), you can easily deduce adjacent values: the previous integer is \(n-1\) and the following one is \(n+1\).
Consecutive integer sequences are prevalent in algebra problems as they hold distinct characteristics that allow easy manipulation and application, such as finding a sum or average of the numbers involved. Understanding them can enhance your ability to solve problems efficiently by recognizing patterns and applying relevant algebraic methods.