数列{an}首项为2,且对任意n∈N*,都有1/a1a2+1/a2a3+...+1/anan+1=n/a1an+1,数列{an}前10项和为110
问题描述:
数列{an}首项为2,且对任意n∈N*,都有1/a1a2+1/a2a3+...+1/anan+1=n/a1an+1,数列{an}前10项和为110
(1)求证:数列{an}为等差数列;
(2)设Cn=an•(1/2)^n,求数列{Cn}的前n项和Tn;
(3)若存在n∈N*,使得an≤(n+1)λ成立,求实数λ的最小值.
答
由题意得1/a1a2+1/a2a3…1/anan-1=(n-1)/a1an①原式-①得1/anan+1=n/a1an+1-(n-1)a1an整理得2=nan-(n-1)an+1两边同时除以n(n-1)得2/n(n-1)=an/(n-1)-an+1/n2/(n-1)-2/n=an/(n-1)-an+1/n(An+1 -2)/n=(an -2)/(n-1)=…...