先观察下列等式，然后用你发现的规律解答下列问题．

$\dfrac{1}{1×2}=1-\dfrac{1}{2}$

$\dfrac{1}{2×3}=\dfrac{1}{2}-\dfrac{1}{3}$

$\dfrac{1}{3×4}=\dfrac{1}{3}-\dfrac{1}{4}$

……

$(1)$计算$\dfrac{1}{1×2}+\dfrac{1}{2×3}+\dfrac{1}{3×4}+\dfrac{1}{4×5}+\dfrac{1}{5×6}=$____；

$(2)$探究$\dfrac{1}{1×2}+\dfrac{1}{2×3}+\dfrac{1}{3×4}+···+\dfrac{1}{n\left(n+1\right)}=$____；$($用含有$n$的式子表示$)$

$(3)$若$\dfrac{1}{1×3}+\dfrac{1}{3×5}+\dfrac{1}{5×7}+···+\dfrac{1}{\left(2n-1\right)\left(2n+1\right)}$的值为$\dfrac{17}{35}$，求$n$的值．