Question

等差数列an中如果存在正整数k和L(k不等于L),使得前k项和Sk=k/L,前l项和SL=L/k,求Sk+L与4的关系,

Answer 1

Sk=ka1+k(k-1)d/2=k/L

a1+(k-1)d/2=1/L (1)

SL=La1+L(L-1)d/2=L/k

a1+(L-1)d/2=1/k (2)

(1)-(2)

(k-L)d/2=1/L-1/k

(k-L)d/2 -(k-L)/kL=0

(k-L)[(d/2)-1/(kL)]=0

已知k≠L,k-L≠0,要等式成立,则d/2=1/(kL)

代入(1)

a1=1/L-(k-1)(d/2)=1/L-(k-1)/(kL)=1/(kL)

S(k+L)=(k+L)a1+(k+L)(k+L-1)(d/2)

=(k+L)/(kL)+(k+L)(k+L-1)/(kL)

=[(k+L)+(k+L)(k+L-1)]/(kL)

=(k+L)(1+k+L-1)/(kL)

=(k+L)²/(kL)

=(k²+2kL+L²)/(kL)

=(k/L)+(L/k)+2

k>0 L>0且k≠L,由均值不等式得k/L+L/k>2

S(k+L)>4

S(k+L)是大于4的.