Question

等差数列am共有2n项,其中奇数项的和为90,偶数项的和为72,且a2n-a1=-33,则该数列的公差为多少?

Answer 1

设等差数列的公差为d,则

a1,a1+d,a1+2d,a1+3d,.,a1+(2n-2)d,a1+(2n-1)d

奇数项:a1,a1+2d,a1+4d,.,a1+(2n-2)d 共n项,公差=2d

偶数项:a1+d,a1+3d,a1+5d,.,a1+(2n-1)d 共n项,公差=2d

奇数项和Sn=na1+n(n-1)*2d/2=na1+n(n-1)d=90 .(1)

偶数项和Tn=n(a1+d)+n(n-1)*2d/2=na1+nd+n(n-1)d=72 .(2)

(2)-(1):nd=-18 ...(3)

a2n-a1=a1+(2n-1)d-a1=(2n-1)d=-33 .(4)

(4)/(3):(2n-1)/n=33/18=11/6

11n=6(2n-1)=12n-6,n=6

d=-18/n=-18/6=-3

所以 d=-3