已知数列\(\{a_{n}\}\)，\(\{b_{n}\}\)满足\(b_{n}=a_{n+1}-a_{n}(n=1,2,3,…)\)．
\((1)\)若\(b_{n}=10-n\)，求\(a_{16}-a_{5}\)的值；
\((2)\)若\(b_{n}=(-1)^{n}(2^{n}+2^{33-n})\)且\(a_{1}=1\)，则数列\(\{a_{2n+1}\}\)中第几项最小？请说明理由；
\((3)\)若\(c_{n}=a_{n}+2a_{n+1}(n=1,2,3,…)\)，求证：“数列\(\{a_{n}\}\)为等差数列”的充分必要条件是“数列\(\{c_{n}\}\)为等差数列且\(b_{n}\leqslant b_{n+1}(n=1,2,3,…)\)”．