Question

已知数列{an}为等差数列,公差d≠0,{an}的部分项组成下列数列：ak1,ak2,…,akn,恰为等比数列,

Answer 1

设公比为q.由已知条件知a1,a3,a11成等比数列.

a3²=a1×a11

(a1+2d)²=a1(a1+10d)

整理,得

2d²-3a1d=0

d(2d-3a1)=0

d=0(与已知矛盾,舍去)或d=3a1/2

a3=a1+2d=a1+3a1=4a1

q=a3/a1=4a1/a1=4

a(k6)=a(k1)q^5

=a1q^5

=a1×4^5

=1024a1

=a1+1023a1

=a1+1023×(2/3)d

=a1+682d

=a1+(683-1)d

k6=683