17. (10 points) Given the quadratic function
$$
f(x)=x^{2}-t x+t(t \in \mathbf{R})
$$

satisfies:
(1) The solution set of the inequality $f(x) \leqslant 0$ has exactly one element; $\square$
(2) If $0<x_{2}<x_{1}$, the inequality $f\left(x_{2}\right)<$ $f\left(x_{1}\right)$ always holds. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=f(n)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Find $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}}$.