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Consecutive integers are integers that follow each ther in order (for example, \(5,6,\) and 7 ). You want to find three consecutive the whose sum is 84 . a. Why does the equation \(n+(n+1)+(n+2)=84\) model the situation? b. Solve the equation in part (a). Then find the three consecutive integers.

Short Answer

Expert verified
The equation \(n+(n+1)+(n+2)=84\) models the situation because it represents the sum of three consecutive integers. The solution to the equation yields \(n=27\). So, the three consecutive integers whose sum is 84 are 27, 28, and 29.

Step by step solution

01

Understand the Model

Consecutive integers continuously follow each other, therefore if one integer is represented by \(n\), the next consecutive integer can be represented by \(n+1\), and the one after that can be represented by \(n+2\). Since the problem states the sum of these three consecutive integers is 84, this situation is modeled by the equation \(n + (n+1) + (n+2) = 84\). The equation represents the sum of three consecutive integers being equal to 84.
02

Simplify the Equation

We can simplify the equation by combining like terms. The left side of the equation \(n + (n+1) + (n+2)\) simplifies to \(3n + 3\). The equation then becomes \(3n + 3 = 84\).
03

Solve for n

To solve for \(n\), first subtract 3 from both sides to isolate the term with \(n\) on one side of the equation. This simplifies the equation to \(3n = 81\). Now, divide both sides by 3 to solve for \(n\). This gives the value of \(n = 27\).
04

Find Three Consecutive Integers

Now that we have found the value of \(n = 27\), we can substitute this value into the expressions \(n\), \(n+1\), and \(n+2\). Thus, the three consecutive integers are 27, 28, and 29.

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