2.
已知数列\(\{a_{n}\}\)为等差数列,\(a_{1}=2\),\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\),数列\(\{b_{n}\}\)为等比数列,且\(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+…+a_{n}b_{n}=(n-1)⋅2^{n+2}+4\)对任意的\(n∈N*\)恒成立.
\((1)\)求数列\(\{a_{n}\}\)、\(\{b_{n}\}\)的通项公式;
\((2)\)是否存在非零整数\(λ\),使不等式\(\sin \dfrac{{a}_{n}π}{4} < \dfrac{1}{λ\left(1- \dfrac{1}{{a}_{1}}\right)\left(1- \dfrac{1}{{a}_{1}}\right)…\left(1- \dfrac{1}{{a}_{n}}\right) \sqrt{{a}_{n}+1}} \)对一切\(n∈N*\)都成立?若存在,求出\(λ\)的值;若不存在,说明理由.
\((3)\)各项均为正整数的无穷等差数列\(\{c_{n}\}\),满足\(c_{39}=a_{1007}\),且存在正整数\(k\),使\(c_{1}\),\(c_{39}\),\(c_{k}\)成等比数列,若数列\(\{c_{n}\}\)的公差为\(d\),求\(d\)的所有可能取值之和.