Question

Insert $x$ arithmetic means between $x^2$ and $1$.

Answer 1

The sequence will from an A.P with

$a=x^2$ and $t_{x+2}=1$

$t_n=a+(n-1)d\Rightarrow t_{x+2}=a+(x+2-1)d\Rightarrow 1=x^2+(x+1)d\Rightarrow {1-x}^2=(x+1)d\Rightarrow (x+1)(x-1)=(x+1)d\Rightarrow d=1-x$
So the arithmetic means are

$a+d,a+2d+a+3d...........a+(x+1)d\Rightarrow x^2-x+1,x^2-2x+2,.......,x$