己知等差数列的前三项依次为a,4,3a,前n项和为Sn,且SK=110,设数列{bn}的通项bn=Sn/n,证明{bn}是等...己知等差数列的前三项依次为a,4,3a,前n项和为Sn,且SK=110,设数列{bn}的通项bn=Sn/n,证明{bn}是等差数列,

Question

己知等差数列的前三项依次为a,4,3a,前n项和为Sn,且SK=110,设数列{bn}的通项bn=Sn/n,证明{bn}是等...
己知等差数列的前三项依次为a,4,3a,前n项和为Sn,且SK=110,设数列{bn}的通项bn=Sn/n,证明{bn}是等差数列,并求其前n项和Tn

Answer 1

2×4=a+3a
4a=8
a=2
4-2=2
数列是以2为首项,2为公差的等差数列.
an=2+2(n-1)=2n
Sn=2(1+2+...+n)=n(n+1)
Sk=2(1+2+...+k)=k(k+1)=110
k^2+k-110=0
(k+11)(k-10)=0
k=10或k=-11(舍去)
bn=n(n+1)/n=n+1
b(n+1)-bn=n+2-(n+1)=1
故{bn}是一个等差数列.其和是Tn=(2+n+1)n/2=n(n+3)/2