Three Consecutive Integers Summing to 51

What You Will Learn

• How to represent consecutive integers using a single variable
• Setting up and solving linear equations from word problems
• The relationship between consecutive integers and their sum
• Why the average of consecutive integers is always the middle value
• Verification strategies for checking integer solutions

Solution: The Variable Setup Method

The key insight is that consecutive integers follow a predictable pattern. If we know the first one, we can express all the others in terms of it.

Step 1 — Define the variable

Let n be the smallest of the three consecutive integers.

Step 2 — Express the three consecutive integers

Since consecutive integers differ by exactly 1, our three integers are:

Step 3 — Set up the equation

The problem states that these three integers add up to 51:

Step 4 — Simplify by combining like terms

Collect all the n terms and all the constants:

Step 5 — Solve for n

Subtract 3 from both sides, then divide by 3:

Step 6 — Find all three integers

Now that we know n = 16, we can find the other two:

Solution: Method 2 — The Average Approach

Here's a completely different way to think about the problem. Instead of starting with the smallest integer, let's use the fact that the sum tells us about the average.

Step 1 — Calculate the average

If three numbers add up to 51, their average is:

Step 2 — Use the consecutive integer property

For any set of consecutive integers, the average is always the middle value. Since our average is 17, the middle integer must be 17.

Step 3 — Find the other two integers

If the middle integer is 17, then:

This gives us the same answer: 16, 17, and 18.

Verification

Let's check our answer by substituting back into the original problem:

Perfect! Our three consecutive integers do indeed add up to 51. We can also verify they're consecutive: 17 = 16 + 1 and 18 = 17 + 1.

Watch Out For These

The Pattern Behind This

Every consecutive integer problem follows the same structure. If you want three consecutive integers with sum S, the formula is:

The middle integer is always S ÷ 3, and the other two are one less and one more than that.

This pattern extends to any number of consecutive integers. For k consecutive integers with sum S, the first integer is always (S - k(k-1)/2) ÷ k, though the average method often works more quickly.

How to Spot This Problem Type

Look for these telltale phrases in word problems:

• "Consecutive integers" or "consecutive whole numbers"
• "Three numbers in a row" or "numbers that follow each other"
• "The next integer" or "the following integer"
• Any problem asking for integers where each is exactly 1 more than the previous

Sometimes these problems are disguised as "page numbers," "house numbers," or "locker numbers" — any sequence where the numbers naturally go in order.