The sum of three consecutive integers is 51. What is the value of the middle integer?

Introduction / Context:
This is a classic problem involving consecutive integers and their sum. Consecutive integers follow one after another without gaps, such as 16, 17, and 18. When their sum is given, you can use simple algebra to find each integer. These questions help reinforce equation setting for simple linear relationships.

Given Data / Assumptions:
- There are three consecutive integers.
- Their sum is 51.
- We are asked to find the middle integer.
- Consecutive integers differ by 1 from each other.

Concept / Approach:
If three consecutive integers are written in terms of a central value, they can be expressed as n - 1, n, and n + 1, where n is the middle number. Their sum then becomes a simple expression in n. Solving this equation gives the value of the middle integer directly. This method avoids having to guess and check values.

Step-by-Step Solution:
Step 1: Let the three consecutive integers be n - 1, n, and n + 1, where n is the middle integer.Step 2: The sum of these integers is (n - 1) + n + (n + 1).Step 3: Simplify the sum: (n - 1) + n + (n + 1) = 3n.Step 4: We are told that this sum equals 51, so 3n = 51.Step 5: Divide both sides by 3 to solve for n: n = 51 / 3 = 17.Step 6: Therefore, the middle integer is 17, and the three integers are 16, 17, and 18.

Verification / Alternative check:
Check the sum of the three consecutive integers found. They are 16, 17, and 18. Their sum is 16 + 17 + 18 = 51, which matches the given total. Since 17 lies exactly between 16 and 18, it is clearly the middle integer. This confirms that our algebraic calculation is consistent with the original problem statement.

Why Other Options Are Wrong:
If the middle integer were 15, the three numbers would be 14, 15, and 16, whose sum is 45, not 51. If it were 16, the sum 15 + 16 + 17 would be 48. A middle integer of 18 would give 17 + 18 + 19 = 54. None of these sums match 51, so they cannot be correct. Only 17 produces the correct total.

Common Pitfalls:
A common mistake is to take 51 and divide by 2 or to think the middle number should be half of the sum. This is incorrect because there are three terms, not two. The correct approach is to represent the three consecutive numbers explicitly in terms of a variable and then use the sum. Remember that for an odd number of consecutive integers, the average of the set will equal the middle integer, which in this case is 51 / 3 = 17.

Final Answer:
The middle integer is 17.