Four Consecutive Integers: Solving for Their Sum

What This Problem Teaches

• Variable definition skills: Learning to represent consecutive integers algebraically using a single variable
• Pattern recognition: Understanding how consecutive quantities relate to each other mathematically
• Equation setup and solving: Translating a word problem into an algebraic equation and solving systematically
• Verification techniques: Checking solutions by substituting back into the original problem conditions
• Integer properties: Recognizing when problems have whole number solutions and understanding why

Solution: Method 1 — The Single Variable Approach

The key insight is to use one variable to represent the first integer, then express all four consecutive integers in terms of that variable.

Step 1 — Define the variable

Let n represent the first (smallest) of the four consecutive integers.

Step 2 — Express all four integers

Since the integers are consecutive, they are:

Step 3 — Set up the equation

The sum of these four integers equals 290:

Step 4 — Simplify the left side

Combine like terms:

Step 5 — Solve for n

Subtract 6 from both sides:

Divide both sides by 4:

Step 6 — Find all four integers

Now that we know n = 71, the four consecutive integers are:

Solution: Method 2 — The Average Method

We can also solve this by thinking about the average of the four consecutive integers.

Step 1 — Find the average

If four integers sum to 290, their average is:

Step 2 — Understand what this means

For four consecutive integers, the average falls exactly between the second and third integers. Since our average is 72.5, the four integers must be centered around this value.

Step 3 — Work outward from the center

If 72.5 is between the second and third integers, then:

Step 4 — Find the remaining integers

Working outward:

So our four consecutive integers are 71, 72, 73, and 74.

Verification

Let's check that our answer is correct by adding the four integers:

We can also verify that these are indeed consecutive integers: each number is exactly 1 more than the previous number.

Additionally, let's check our algebra by substituting n = 71 back into our original equation:

Common Pitfalls

Here are the mistakes students most frequently make with consecutive integer problems:

The General Formula

For any set of k consecutive integers starting with n, the sum follows this pattern:

For four consecutive integers specifically, this simplifies to:

This formula explains why consecutive integer problems always have clean solutions when the given sum has the right remainder when divided by the count of integers. In our case, 290 ÷ 4 = 72 remainder 2, and that remainder of 2 is exactly what we expect from the "+6" term after dividing by 4.

How to Spot This Problem Type

Look for these key phrases that signal a consecutive integer problem:

• "consecutive integers" — the most obvious signal
• "integers in a row" — another way of saying consecutive
• "three numbers that follow each other" — describes the consecutive relationship
• "even consecutive integers" — like 2, 4, 6, 8 (every other even number)
• "odd consecutive integers" — like 3, 5, 7, 9 (every other odd number)

The structure is always: some number of integers that follow a pattern, and you're given information about their sum, difference, or product.

What If?