Problem 7 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Consecutive Integers

Let's identify the three consecutive integers. If the first integer is represented by the variable \(n\), then the next two consecutive integers would be \(n+1\) and \(n+2\). This is because consecutive integers increase by 1.

Step 2: Write the Expression for the Sum

Now that we have the three integers as \(n\), \(n+1\), and \(n+2\), we express their sum as a mathematical equation: \(n + (n + 1) + (n + 2)\).

Step 3: Simplify the Expression

Combine the terms in the equation from Step 2. This is done by adding together the like terms: \(n + n + 1 + n + 2 = 3n + 3\).

Step 4: Final Expression

The sum of the three consecutive integers can be expressed as \(3n + 3\), where \(n\) is the first integer.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order without gaps, typically increasing by 1 as you move from one to the next. An easy way to think about them is using whole numbers like 3, 4, and 5, which are consecutive integers. When they're connected to algebra, they usually require you to understand how to represent these integers using a variable.
In many math problems, we might express the first integer with a variable like "n." Therefore, the next consecutive integers would become "n+1" and "n+2." This helps in writing algebraic equations involving these integers easily. The pattern goes on, which means, if you want more consecutive numbers, they would simply follow as "n+3," "n+4," and so forth.
This kind of representation is very useful in different problems, especially those that require calculating sums or differences of consecutive numbers.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. These can include operations like addition, subtraction, multiplication, and division. They are the building blocks of algebra used to make complex equations more manageable.
In the context of our problem with consecutive integers, the algebraic expression we work with is a representation of those integers as a sum:

• The first integer is "n."
• The second integer, one more than the first, is "n+1."
• The third is "n+2."

When you sum them up, it results in the expression: \[ n + (n+1) + (n+2) \] This simple equation represents the addition of all three integers we've defined with algebraic language, setting the stage for further simplification.

Simplifying Expressions

Simplifying an expression means reducing it to its most basic form without changing its value. It's an essential part of solving algebraic problems, allowing you to see and understand the core relationship between the variables involved.
In our example, the sum of consecutive integers starts as \[ n + (n+1) + (n+2) \]. To simplify, you combine like terms, which are those that contain the same variable or are constant terms. Here, you combine the "n" terms and the constant numbers:

• Combine all "n" terms: \[ n+n+n = 3n \]
• Combine all constant terms: \[ 1 + 2 = 3 \]

After combining these, the expression simplifies to \[ 3n + 3 \]. This expression gives a cleaner form, highlighting the relationship between the integers where "n" is the first integer. Simplifying expressions provides a clearer view of the mathematical relationships and makes calculations easier.