Question

公差不为零的等差数列｛an｝的前n项和为Sn，若a4是a3与a7的等比中项，S8＝32，则S10等

公差不为零的等差数列｛an｝的前n项和为Sn，若a4是a3与a7的等比中项，S8＝32，则S10等于？

Answer 1

解：
设公差为d，则d≠0。
a4是a3、a7的等比中项，则
a4²=a3·a7
(a3+d)²=a3·(a3+4d)
整理，得2a3d-d²=0
d(2a3-d)=0
d=0(舍去)或2a3-d=0
a3=d/2
a1=a3-2d=d/2 -2d=(-3/2)d
S8=8a1+8×7d/2=8·(-3/2)d+28d=16d=32
d=2
a1=(-3/2)d=(-3/2)·2=-3
S10=10a1+10×9d/2=10×(-3)+10×9×2/2=60

Answer 2

an = a1+(n-1)d
S8＝32
4(2a1+7d) = 32
2a1+7d = 8
a1 + (7/2)d = 4 (1)
a4是a3与a7的等比中项
a3.a7=(a4)^2
(a1+2d)(a1+6d) = (a1+3d)^2
(4-(3/2)d)( 4 + (5/2)d ) = ( 4 -(1/2)d )^2
(8-3d)( 8 + 5d ) = ( 8 -d )^2
16d -15d^2 = -16d + d^2
16d^2 -32d=0
d=2
from (1)
a1 + (7/2)d = 4
a1+7=4
a1=-3
an = -3+2(n-1) = 2n -5
S10 = 5(a1+a10) = 5(-3 +15) =60

Answer 3

直接用求前n项和的公式和等差数列及等比数列公式求解不就得了