Question

已知数列an为等差数列,若a11/a10<-1,且它们的前n项和有最大值,则使得Sn<0的n最小值是多少?

Answer 1

Sn有最大值，说明d<0，且a1>0.
a11/a10 < -1 => a11 < 0 且，a10 > 0, |a11| > |a10|.
a11 = a1 + 10d < 0, =>2a1 + 20d < 0
a10 = a1 + 9d > 0. =>2a1 + 18d > 0
=>a10 + a11 = 2a1 + 19d < 0
Sn = n*a1 + n(n-1)*d/2 < 0
=>a1 + (n-1)*d/2 < 0
=>2a1 + (n-1)*d <0
以上，可知最小n为Nmin = 19+1 = 20
且有
S11 = S9 + a10 + a11 <S9
S9 = S10 - a10 < S10, S10为最和最大值。