# Sum of Consecutive Integers

When solving word problems involving consecutive integers, it’s important to remember that we are looking for integers that are one unit apart.

So if we have n as the first integer, then n + 1 will be the second integer, n + 2 will be the third integer, n + 3 will be the fourth, n + 4 will be the fifth, and so on.

Let’s take for example: 15, \left( {15 + 1} \right), \left( {15 + 2} \right), \left( {15 + 3} \right), \left( {15 + 4} \right)

Our results come up to: 15,\,16,\,17,\,18,\,19

When dealing with consecutive integers, notice that the difference between the larger and smaller integers is always equal to 1.

**Observe the following:**

- \left( {n + 1} \right) - \left( n \right) = n + 1 - n = n - n + 1 = 1

- \left( {n + 2} \right) - \left( {n + 1} \right) = n + 2 - n - 1 = n - n + 2 - 1 = 1

- \left( {n + 3} \right) - \left( {n + 2} \right) = n + 3 - n - 3 = n - n + 3 - 2 = 1

- \left( {n + 4} \right) - \left( {n + 3} \right) = n + 4 - n - 3 = n - n + 4 - 3 = 1

## Examples of Solving the Sum of Consecutive Integers

**Example 1:** The sum of three consecutive integers is 84. Find the three consecutive integers.

The first step to solving word problems is to find out what pieces of information are available to you.

For this problem, the following facts are given:

- We need to ADD
**three**integers that are consecutive - The numbers are
**one unit apart**from each other - Each number is
**one more**than the previous number - The
**sum**of the consecutive integers is 84

With these facts at hand, we can now set up to represent our three consecutive integers.

Let n be our **first integer**. Therefore,

We’re now ready to write our equation. Remember that we are asked to find the sum, so we will be adding our three consecutive integers.

Let’s proceed and solve the equation.

Now that we have the value for the variable “n, we can use this to identify the three consecutive integers.

Finally, let’s do a quick check to make sure that the sum of the consecutive integers 27, 28, 29 is indeed 84 as given in our original problem.

**Example 2**: Find four consecutive integers whose sum is 238.

To start, let’s go ahead and determine the important facts that are given in this problem.

- We will be
**adding**four integers that follow one another - The integers are one unit apart
- The
**sum**of the four consecutive integers is 238

The next step is to represent the four consecutive integers using the variable “n“.

Let n be the **first integer**. Since the four integers are consecutive, this means that the **second integer** is the * first integer increased by 1* or {n + 1}. In the same manner, the

**third integer**can be represented as {n + 2} and the

**fourth integer**as {n + 3}.

We can then translate “the sum of four consecutive integers is 238” into an equation.

Solve the equation:

At this point, the value of n is not our final answer. We will use this value, however, to find the other three integers that follow after n which is 58.

The last condition that we need to satisfy is that the sum of the four consecutive integers must be 238. Obviously, since “sum” means to add, we will be adding the integers 58, 59, 60, and 61.

**Example 3**: The sum of six consecutive integers is - \,9. What are the integers?

*What do we know?*

- We will
**add**six consecutive integers - The
**sum**of the integers is - \,9 - The integers will be
**one unit apart**but will more likely involve**negative integers**

**Represent the six consecutive integers**. This time, we’ll use x as our variable.

Let x be the **1st integer**.

Next, let’s **translate **“the sum of six consecutive integers is - \,9“** into an equation**.

x + \left( {x + 1} \right) + \left( {x + 2} \right) + \left( {x + 3} \right) + \left( {x + 4} \right) + \left( {x + 5} \right) = - \,9

**Solve**:

Now that we know the value of x which is - \,4, let’s **determine** **the six consecutive integers**.