The sum of three consecutive even numbers is 72 — find them

Question

The sum of three consecutive even numbers is 72. Find the numbers.

Solution — Step by Step

Consecutive even numbers differ by 2. Let the smallest of the three be $x$ .

Then the three consecutive even numbers are: $x$ , $x+2$ , $x+4$ .

x + (x+2) + (x+4) = 72

3x + 6 = 72

3x = 66

x = 22

The three consecutive even numbers are:

22, \quad 24, \quad 26

Verification: $22 + 24 + 26 = 72$ ✓

Why This Works

Even numbers always differ by 2, so “consecutive even numbers” means we add 2 each time. Writing them as $x$ , $x+2$ , $x+4$ captures this pattern with just one unknown. The sum gives a simple linear equation with one unknown.

Alternative Method — Using the Middle Number

The three numbers are equally spaced (by 2). The middle number is the average:

\text{Middle number} = \frac{72}{3} = 24

The three numbers: $24-2 = 22$ , $24$ , $24+2 = 26$ .

This shortcut (middle number = average) works for any three consecutive numbers (even, odd, or integers) because the terms are in arithmetic progression with the middle term as the mean.

Common Mistake

Students sometimes represent consecutive even numbers as $x$ , $x+1$ , $x+2$ — those are consecutive integers, not consecutive even numbers. Consecutive even numbers always differ by 2: use $x$ , $x+2$ , $x+4$ . Similarly, consecutive odd numbers also differ by 2: $x$ , $x+2$ , $x+4$ (where $x$ is odd).

A quick sanity check for this type of problem: the three numbers should be close together, near the average (72÷3 = 24). If your answer gives numbers far from 24 or non-even numbers, recheck the setup.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using the Middle Number

Common Mistake

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