Prove that the mean of any four consecutive even integers is an integer. - OCR - GCSE Maths - Question 13 - 2019 - Paper 6

Let the four consecutive even integers be represented as:

• First integer: $2n$
• Second integer: $2n + 2$
• Third integer: $2n + 4$
• Fourth integer: $2n + 6$

where $n$ is an integer.

To find the mean, calculate the sum of these integers and divide by the number of integers (which is 4):

1. Sum of the integers:
   $ext{Sum} = 2n + (2n + 2) + (2n + 4) + (2n + 6)$
   Simplifying this gives:
   $ext{Sum} = 2n + 2n + 2 + 2n + 4 + 2n + 6 = 8n + 12$

2. Mean Calculation:
   $ext{Mean} = \frac{8n + 12}{4} = 2n + 3$

Since both $2n$ and 3 are integers, their sum, $2n + 3$, must also be an integer.

Thus, we have proved that the mean of any four consecutive even integers is indeed an integer.