Question

若等差数列{a_n}中，

lim

n→∞

n(an+n)

Sn+n

=1，则公差d=______．

Answer 1

等差数列{a_n}中，a_n=a₁+（n-1）d，S_n=na₁+

n(n-1)

2

d，
所以

n(an+n)

Sn+n

=

n(a1+nd+n-d)

na1+n + n(n-1) 2 d

=

2(a1+nd+n-d)

2a1+2 +(n-1)d

；

lim

n→∞

n(an+n)

Sn+n

=

lim

n→∞

 

2(a1+nd+n-d)

2a1+2 +(n-1)d

=

lim

n→∞

2a1-2d n +2d+2

2a1+2-d m +d

=

2d+2

d

=1
d=-2．
故答案为：-2．