Question

设S_n=1+2+3+…+n，n∈N^*，则函数f(n)=

Sn

(n+32)Sn+1

的最大值为______．

Answer 1

由题意S_n=1+2+3+…+n=

n(n+1)

2

∴f(n)=

Sn

(n+32)Sn+1

=

n(n+1) 2

(n+32)  × (n+2)(n+1) 2

=

n

(n+32)  ×(n+2)

=

1

n+34+ 64 n

≤

1

34+16

=

1

50

等号当且仅当n=

64

n

=8时成立
故答案为

1

50