已知a>0,a≠1,an是首项与公比为a的等比数列,bn满足bn=anlgan①求an并证明数列{an}是等比数列 ②求数列{bn}的前n项和Tn③若对一切n∈N正都有bn

问题描述:

已知a>0,a≠1,an是首项与公比为a的等比数列,bn满足bn=anlgan
①求an并证明数列{an}是等比数列
②求数列{bn}的前n项和Tn
③若对一切n∈N正都有bn

an=a^n……(证明数列{an}是等比数列是多此一举,已知中有条件)
bn=a^n*lg(a^n)
 =(lga)*n*a^n
设cn=n*a^n
Scn   =a+2*a^2+3*a^3+4*a^4+………+(n-1)*a^(n-1)+n*a^n
a*Scn  =  a^2+2*a^3+3*a^4+4*a^5+………………+(n-1)*a^n+n*a^(n+1)
a*Scn-Scn=n*a^(n+1)-a-(a^2+a^3+……a^n)
     =n*a^(n+1)-a-[a^2*(a^(n-1)-1)]/(a-1)
     =n*a^(n+1)-[a^(n+1)-a)]/(a-1)
Scn   =[(a-1)*n*a^(n+1)-a^(n+1)+a]/(a-1)^2
Tn    =(lga)*[(n*a-n-1)*a^(n+1)+a]/(a-1)^2
bn(lga)*n*a^n(lga)*[n-(n+1)*a]a0(n∈N+)或a>1且n-(n+1)*aa1且a>n/(n+1)(n∈N+)
a1且a>=1
故01

①an=a*a^(n-1)=a^n②bn=anlgan=nlga*a^nTn=lga*a+lga*2a²+lga*3a³+...+lga*na^n ...①aTn=lga*a²+lga*2a³+...+lga*na^(n+1) ...②由①,②得(1-a)Tn=alga+a²lga+a³lga+a^n *lga -lga*...