8．
平面直角坐标系\(xoy\)中，已知点\(\left(n,{a}_{n}\right) n∈{N}^{*} \)在函数\(y={a}^{x}\left(a\geqslant 2,a∈N\right) \)的图像上，点\(\left(n,{b}_{n}\right) (n∈{N}^{*} )\)在直线\(y=\left(a+1\right)x+b b∈R \)上\(.\)
\((1)\)若点\(\left(1,{a}_{1}\right) \)与点\(\left(1,{b}_{1}\right) \)重合，且\({a}_{2} < {b}_{2} \)，求数列\(\left\{{b}_{n}\right\} \)的通项公式；
\((2)\)证明：当\(a=2\)时，数列\(\left\{{a}_{n}\right\} \)中任意三项都不能构成等差数列；
\((3)\)当\(b=1\)时，记\(A=\left\{ \left.x \right|x={a}_{n},n∈{N}^{*}\right\} \)，\(B=\left\{ \left.x \right|x={b}_{n},n∈{N}^{*}\right\} \)，设\(C=A∩B \)，将集合\(C\)的元素按从小到大的顺序排列组成数列\(\left\{{c}_{n}\right\} \)，写出数列\(\left\{{c}_{n}\right\} \)的通项公式\({c}_{n} \)．