Problem 8 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The sum is \(3n\).

Step 1: Understanding the Problem

We need to find the sum of three consecutive integers where the middle integer is given as \( n \). We will define the other two integers in relation to \( n \) and then find their sum.

Step 2: Defining the Integers

Three consecutive integers can be written in terms of \( n \) as follows: the first integer is \( n-1 \), the second integer is \( n \), and the third integer is \( n+1 \).

Step 3: Setting Up the Expression

The sum of these three consecutive integers is given by \((n-1) + n + (n+1)\).

Step 4: Combining Like Terms

Combine the terms in the expression: \((n-1) + n + (n+1) = n - 1 + n + n + 1\).

Step 5: Simplifying the Expression

Simplify the expression by combining like terms: \(3n\). Thus, the sum of the three consecutive integers is \(3n\).

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow one another without any gaps. When you hear about consecutive numbers, think about counting numbers like 1, 2, 3, or even 10, 11, 12. They increase or decrease by one each time. This concept is especially useful in algebra problems when trying to find unknown values by using known numbers and relationships.

• The smallest integer in the set is often represented as n−1.
• The middle integer is just n.
• The largest integer would be n+1.

Knowing how to express consecutive integers algebraically is crucial for solving various math problems involving their sum, differences, or products. It helps to simplify and create equations that reveal complex relationships.

Integer Sum

The sum of integers can tell you a lot about the numbers involved. In our exercise, we consider three consecutive integers, which are n−1, n, and n+1.The important thing to remember is that when adding these integers, certain things will always happen. You will find that:- The term n appears three times when you write the equation - The constants that arise from the first and last integers, −1 and +1, will cancel each other outGiven these three integers, their sum is \[(n-1) + n + (n+1) = n - 1 + n + n + 1.\] You'll notice the -1 and +1 vanish as they sum to zero. This leaves you with \[ 3n, \] meaning any set of three consecutive integers will have a sum equal to three times the middle integer.

Algebra Problem Solving

Solving algebra problems requires a methodical approach, often starting with understanding what the problem asks. For algebra involving numbers, like consecutive integers, setting out clear steps will always help you get to the solution.When tackling these exercises, remember to:

• Define your variables clearly. Here, we used n to represent the middle integer.
• Use known relationships to express other numbers in terms of these variables. This step involved expressing the consecutive integers as n−1, n, and n+1.
• Set up the equation by combining the variables logically, simplifying whenever possible. We set up the sum as \((n-1) + n + (n+1)\) and then simplified.
• Simplify to find the answer. This exercise showed how like terms could be combined to simplify to 3n.

These steps form the core of solving many algebraic problems. By practicing this method, you'll be equipped to tackle various mathematical exercises with confidence.