The Sum of Three Consecutive Even Integers Is 102. What Is the Largest Integer?

Finding the largest of three consecutive even integers that add up to 102 can be a fun and rewarding math puzzle—whether you're a student learning algebra, a teacher illustrating number patterns, or someone who simply enjoys practicing logic. In this article, we’ll walk through the steps to solve this problem clearly and confidently.

Understanding Consecutive Even Integers

Understanding the Context

Consecutive even integers are numbers that follow one after the other in the sequence of even numbers, increasing by 2. For example, if one integer is n, then the next two consecutive even integers are n + 2 and n + 4.

We are told the sum of three such consecutive even integers equals 102:
n + (n + 2) + (n + 4) = 102

Step-by-Step Equation Solution

  1. Combine like terms:
    Add the expressions:
    n + (n + 2) + (n + 4) = 3n + 6
Solved The sum of three consecutive even integers is 240 | Chegg.com

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Solved The sum of three consecutive even integers is 240 | Chegg.com
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Consecutive Integers - Math Monks
Solved c. The sum of three consecutive integers if the | Chegg.com
SOLVED: There are three consecutive Integers such that the sum of one ...

Key Insights

So, the equation becomes:
3n + 6 = 102

  1. Solve for n:
    Subtract 6 from both sides:
    3n = 96

Divide both sides by 3:
n = 32

  1. Find the three integers:
    The middle number is 32. Therefore:
    - First even integer: 32
    - Second even integer: 32 + 2 = 34
    - Third (largest) even integer: 32 + 4 = 36

Verify the Sum

Continue Reading

Alternative: in optics, spherical surface with radius R, when light enters normally, no angular shift but refraction at exit. But curves after passing suggests spherical refraction.
Standard: for a spherical surface entering normally, no deflection. But if curved exit surface, yet dome is uniform, perhaps model as plano-convex?
But given: focal spot 200 m beyond center — total 50 + 200 = 250 m from entry center?

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Final Thoughts

Let’s check:
32 + 34 + 36 = 102 ✅
That confirms our solution is correct.

Final Answer

The largest of the three consecutive even integers is 36.

This simple yet powerful problem reinforces basic algebraic thinking, helps identify patterns in number sequences, and demonstrates how to translate words into equations. Next time you encounter three consecutive even integers summing to a known value, you’ll know exactly how to find the largest with ease!