Solving for Consecutive Even Numbers with a Median of 31

The problem at hand involves finding four consecutive even numbers where the median is 31. This can be approached through multiple methods, each providing a unique insight and verification of the solution. Let's explore these different approaches and the underlying mathematical principles.

Brute Force Solution using J Programming Language

The simplest approach is through a brute force method using the J programming language. The J language is known for its concise syntax and powerful mathematical capabilities, making it an ideal tool for such problems. Here is a step-by-step breakdown:

Initialization: We start by initializing the variable n with the median value of 31.

Calculate: Using the median function in J, we find the numbers that would result in a median of 31.

Output: The numbers that satisfy the condition are obtained and displayed.

Here is the J code: n ~ 31 median; 28 30 32 34.

Algebraic Solution

A more analytical approach is to use basic algebra. Let's denote the smallest of the four consecutive even numbers as x. The sequence would then be x, x 2, x 4, x 6. The median of this sequence is the average of the two middle numbers, which is (x 2 x 4) / 2 (2x 6) / 2 x 3.

Given that the median is 31, we set up the equation:

x 3 31

Solving for x, we get:

x 28

Therefore, the four consecutive even numbers are 28, 30, 32, and 34.

Direct Method with Summation

An even more straightforward method involves using the sum of the sequence. The sum of the four consecutive even numbers is:

4x 12 124

Here, 4x represents the sum of the four numbers, and 12 is the sum of the even differences (2 4 6).

Solving the equation:

4x 12 124

4x 112

x 28

Thus, the numbers are 28, 30, 32, and 34.

Theoretical Explanation with Median Calculation

To further solidify our understanding, let's verify the solution using the theoretical properties of the median. For a set of consecutive even numbers, the median is the average of the two middle numbers.

Let the numbers be 2k, 2k 2, 2k 4, 2k 6. The median is:

(2k 2 2k 4) / 2 2k 3

Given that the median is 31, we set:

2k 3 31

Thus:

2k 3 31

2k 28

k 14

Substituting k back into the sequence, we get:

2 * 14 28, 2 * 14 2 30, 2 * 14 4 32, 2 * 14 6 34

Therefore, the numbers are 28, 30, 32, and 34.

Conclusion

In conclusion, the four consecutive even numbers with a median of 31 are 28, 30, 32, and 34. This solution can be confirmed through various methods, each offering a unique perspective on solving the problem.