已知数列{a n}是等差数列,Sn是其前n项和,求证：S5,S10-S5,S15-S10这三个数也成等差数列.

Question

已知数列{a n}是等差数列,Sn是其前n项和,求证：S5,S10-S5,S15-S10这三个数也成等差数列.

Answer 1

s5=(a1+a5)*5/2=(a1+a1+4d)*5/2=5(a1+2d)=5a1+10d
s10=(a1+a10)*10/2=(a1+a1+9d)*5/2=5(2a1+9d)=10a1+45d
s15==(a1+a15)*15/2=(a1+a1+14d)*15/2=15(a1+7d)=15a1+105d
s10-s5=10a1+45d-5a1-10d=5a1+35d
s15-s10=15a1+105d-10a1-45d=5a1+60d
s10-s5-s5=5a1+35d-5a1-10d=25d
s15-s10-(s10-s5)=5a1+60d-5a1-35d=25d
所以S5,S10-S5,S15-S10这三个数也成等差数列 公差为25d