In $\mathbb{R}^2$, a circle of radius $r$, centered at $(a_1,a_2)$, is represented by the equation $(x_1-a_1)^2+(x_2-a_2)^2=r^2$.
In $\mathbb{R}^3$, a sphere of radius $r$, centered at $(a_1,a_2,a_3)$, is represented by the equation $(x_1-a_1)^2+(x_2-a_2)^2+(x_3-a_3)^2=r^2$.
Does the equation $(x_1-a_1)^2+(x_2-a_2)^2+ \cdot \cdot \cdot + (x_n-a_n)^2 = r^2$ represent a "sphere" in $\mathbb{R}^n$ of radius $r$ and centered at $(a_1,a_2,...,a_n)$?
Generally if we have a normed space $(E, ||\cdot||)$, then the sphere of radius $r>0$ and center $a\in E$ is defined as the set $$\{x\in E: ||x-a||=r\}.$$ In the euclidean space $\mathbb{R}^{n+1}$ with the usual norm, this set is defined by the equation $$\sum_{i=1}^{n+1}(x_i-a_i)^2=r^2.$$