已知{an}为递减的等比数列,且{a1,a2,a3}⊊{-4,-3,-2,0,1,2,3,4}. (Ⅰ)求数列{an}的通项公式; (Ⅱ)当bn=1−(−1)n2an时,求证:b1+b2+b3+…+b2n−1
问题描述:
已知{an}为递减的等比数列,且{a1,a2,a3}⊊{-4,-3,-2,0,1,2,3,4}.
(Ⅰ)求数列{an}的通项公式;
(Ⅱ)当bn=
an时,求证:b1+b2+b3+…+b2n−1<1−(−1)n
2
. 16 3
答
(Ⅰ)∵{an}是递减数列,∴数列{an}的公比q是正数,∵{a1,a2,a3}⊊{-4,-3,-2,0,1,2,3,4},∴a1=4,a2=2,a3=1,∴q=a2a1=14=12,∴an=a1qn−1=82n.(Ⅱ)由(1)得,bn=1−(−1)n2an=8[1−(−1)n]...