Question

数列an是公差不为0等差数列,a2=2,a3、a6、a12等比,则1/a1a3+1/a2a4+……1/an.a+2=?

Answer 1

an=a1+(n-1)d
a2=a1+d=2,a1=2-d
a3=a1+2d=2+d,
a6=a1+5d=2+4d,
a12=a1+11d=2+10d
a3a6=2a12
(2+d)(2+4d)=2(2+10d)
(2+d)(1+2d)=2+10d
2d^2=5d
d=5/2
an=5n/2-3
ana(n+2)=[(5n-6)/2][(5n+4)/2]
1/[ana(n+2)]=4/[(5n-6)(5n+4)]=(2/5)[1/(5n-6)-1/(5n+4)]
(5/2)T=(5/2){1/(a1a3)+1/(a2a4)+……+1/[ana(n+2)]}
=[1/(-1)-1/9]+[1/4-1/14]+[1/9-1/19]+[1/14-1/24]+……+[1/(5n-21)-1/(5n-11)]+[1/(5n-16)-1/(5n-6)]+[1/(5n-11)-1/(5n-1)]+[1/(5n-6)-1/(5n+4)]
=[1/(-1)]+[1/4]+[-1/(5n-1)]+[-1/(5n+4)]
=-3/4-1/(5n-1)-1/(5n+4)
(5/2)T=-3/4-1/(5n-1)-1/(5n+4)
=-3/4-(10n+3)/[(5n-1)(5n+4)]
T=-3/10-2(10n+3)/[(25n-5)(5n+4)]