Find the maximal constant M, such that for arbitrary integer n≥3, there exist two sequences of positive real number a1,a2,⋯,an, and b1,b2,⋯,bn, satisfying
(1): \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;
(2): a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M. inequalitiesgeometric sequencealgebra proposedalgebra