PLEASE HELP ME! WILL MARK BRAINLIEST! I WILL REPORT ABSURD AND INCORRECT ANSWERS!
2. Suppose, 11 numbers are given. When each of them was increased by 1, the sum of their squares did not change. Once again each number is increased by 1. Does the sum of squares change this time, and if so, by how much?
4. Several points were marked on a line after which a point was added between any two neighboring points. This operation was repeated 2 more times, and as a result, there were 65 points on the straight line. How many points were there in the beginning?

[tex]x_1^2+...+x_{11}^2=(x_1+1)^2+...+(x_{11}+1)^2\\\\=x_1^2+...+x_{11}^2+2(x_1+...+x_{11})+11\\\\\text{So } 2(x_1+...+x_{11})+11 = 0\\\\(x_1+2)^2+...+(x_{11}+2)^2=x_1^2+...+x_{11}^2+4(x_1+...+x_{11})+4*11\\\\\text{As we know that }2(x_1+...+x_{11})+11 = 0\\4(x_1+...+x_{11})+4*11=-11*2+4*11=22[/tex]

So the sum of squares changes by

[tex]\boxed{ \ 22 \ }[/tex]

PART 4

Let's say that we have n points at the beginning.

We will add n-1 points the first time, we will get n + n - 1 = 2n -1 points.

And then, the second time we add 2n - 1 - 1 points and we get

2n - 1 +2n - 2 = 4n - 3.

Finally, we do it a last time, we add 4n - 3 - 1 points and we get 4n - 3 + 4n - 4 points = 8n - 7 and it must be 65.

So, 8n -7 = 65 <=> 8n = 65+7=72 <=> n = 72/8=9

[tex]\boxed{= \ 9}[/tex]

Thank you