等差数列{an}中,a3=8,a7=20.若数列{1/an*an+1}的前n项和为4/25.则n的值为多少?地区统考的卷子。应该不会出这么大岔子。

Question

等差数列{an}中,a3=8,a7=20.
若数列{1/an*an+1}的前n项和为4/25.
则n 的值为多少?
地区统考的卷子。应该不会出这么大岔子。

Answer 1

a3=a+2d=8
a7=a+6d=20
d=3,a=2
1/[an*a(n+1)]=1/{[a+(n-1)d][a+nd]}
=1/[(3n-1)(3n+2)]
=(1/3)[1/(3n-1)-1/(3n+2)]
Tn=(1/3){[1/2-1/5]+[1/5-1/8]+[1/8-1/11]+……+[1/(3n-7)-1/(3n-4)]+[1/(3n-4)-1/(3n-1)]+[1/(3n-1)-1/(3n+2)]}
=(1/3)[1/2-1/(3n+2)]
=4/25
(1/3)[1/2-1/(3n+2)]=4/25
1/(3n+2)=1/2-12/25=1/50
(3n+2)=50
n=16