The sum of three consecutive even integers is 90. What is the largest of these integers? - Coaching Toolbox
Understanding the Sum of Three Consecutive Even Integers: What Is the Largest?
Understanding the Sum of Three Consecutive Even Integers: What Is the Largest?
When you encounter a problem like “The sum of three consecutive even integers is 90,” it’s more than just a math puzzle—it’s a gateway to understanding sequences, algebra, and logical reasoning. In this article, we’ll solve the equation step-by-step and reveal how to find the largest of these integers. Whether you're a student, a teacher, or someone curious about number patterns, this is a clear and practical guide.
Understanding the Context
What Are Consecutive Even Integers?
Consecutive even integers are numbers like 2, 4, 6 or 14, 16, 18 — each differing by exactly two. Because even numbers are always two units apart, finding three in sequence is straightforward: if the first is n, the next two are n+2 and n+4.
Setting Up the Equation
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Key Insights
Let the first even integer be:
n
Then the next two consecutive even integers are:
n + 2 and n + 4
The sum of these three integers is given as 90:
n + (n + 2) + (n + 4) = 90
Solving the Equation Step-by-Step
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Combine like terms:
3n + 6 = 90 -
Subtract 6 from both sides:
3n = 84 -
Divide both sides by 3:
n = 28
Identifying All Three Integers
Now that we know n = 28, the three consecutive even integers are:
- First: 28
- Second: 30
- Third (largest): 32
Verifying the Solution
Let’s check:
28 + 30 + 32 = 90 ✅
All numbers are even and consecutive in sequence ✅
The largest is indeed 32 ✔️
