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

The sum of three consecutive even integers is 90. What is the largest of these integers?

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

Continue Reading

The circle has radius 8, ellipse semi-minor axis is 8 — so the circle touches the ellipse at the top and bottom.
But does it intersect in more points? No — since the circle lies entirely on or inside the ellipse in \( y \), but at \( x=0 \), both are 1.
Actually, the circle is inscribed: at \( x=0 \), both have \( y = \pm8 \), and for \( |x| > 0 \), ellipse allows larger \( x \), but circle restricts \( x^2 + y^2 = 64 \), so maximum \( y \) is bounded.

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Final Thoughts

  1. Combine like terms:
    3n + 6 = 90

  2. Subtract 6 from both sides:
    3n = 84

  3. 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 ✔️