Question

等差数列{an}中,a1=1,前n项和为Sn,公差为d,若S2,S3-S1,S5-S3成等比数列

Answer 1

(1)

an=1+(n-1)d

S2=a1+a2=2+d

S3-S1=a2+a3=2+3d

S5-S3=a4+a5=2+7d

(2+3d)^2=(2+d)(2+7d)

d=0 or 2

d=2时

bn=1/an*an+1=(1/an-1/an+1)/(an+1-an)=(1/an-1/an+1)/d

an=2n-1

b1=(1/a1-1/a2)/2

b2=(1/a2-1/a3)/2

...

bn=(1/an-1/an+1)/2

b1+b2+..bn=(1/a1-1/a2+1/a2-1/a3+...+1/an-1/an+1)/2

Tn=(1/a1-1/an+1)/2=(1-1/2n+1)/2=n/(2n+1)

d=0时

an=1

bn=1

Tn=n

(2)kTn=nk/(2n+1)