Proof for positive definiteness of a given matrix

Question

Let A be N$\times$ N symmetric matrix having all real entries. The entries of A be $a_{ij}$ where $i,j \in {1,2,..N}$ and $a_{ij}$ is given by $$a_{ij}=\begin{cases} f(n)=\dfrac{n^\zeta-2(n-1)^\zeta+(n-2)^\zeta}{2}, & \text{for } i\neq j, \text{$n= |i-j+1|$} \\ 1, & \text{for } i=j \\ \end{cases}$$ Where $1\leq \zeta \leq 2$. Prove that A is a positive definite matrix.