Question

在等差数列{an}中,已知a1=20,前n项和为Sn,且S10=S15

当n为何值时,Sn有最大值?并求出它的最大值

Answer 1

S10=10*a1+10*9/2*d=200+45d
S15=15*a1+15*14/2*d=300+105d
200+45d=300+105d
解得d=-5/3
Sn=na1+n(n-1)d/2=20n-5/6n(n-1)=-5/6n^2+(125/6)n
当n=(x1+x2)/2=25时，Sn有最大值
n=25时，Sn=-5/6n^2+125/6*n=0

Answer 2

an = 20+(n-1)d
S10=S15
(40+9d)5 = (20+7d)15
40+9d=60+21d
12d=-20
d= -5/3
an = 20-(5/3)(n-1)
a13=0
max Sn when n=12 or 13
S13= (20)13/2 = 130