## Task 1 - 270811

Steffen presents the following task to the members of the AG Mathematics group:

"Each of you think of a number, multiply this number by 2, add 30 to the product, divide the sum by 2, and subtract the initially thought number from the obtained result!

Write down the result!"

It turned out that all students in the working group had the same result. Must Steffen's classmates have necessarily thought of the same number?
The students do not necessarily have to think of the same number. It is observed that two different initial numbers result in the same outcome. For example, the following observations suffice as proof:

The initial number 1 leads successively to the numbers $1 \cdot 2=2, 2+30=32, 32: 2=16, 16-1=15$. The initial number 2 leads successively to the numbers $2 \cdot 2=4, 4+30=34, 34: 2=17, 17-2=15$.

Another proof possibility: One proves that from any initial number $a$, the same result is obtained, namely via the numbers

$$
a \cdot 2 ; \quad a \cdot 2+30 ; \quad (a \cdot 2+30): 2=a+15 ; \quad a+15-a=15
$$