等比数列An中,若Sn=12,S2n=24,求S3n

等比数列An中,若Sn=12,S2n=24,求S3n
等比数列An中,若Sn=12,S2n=24,求S3n

等比数列An中,若Sn=12,S2n=24,求S3n
a(n)=aq^(n-1),
q=1时,s(n) = na.
12 = s(n) = na,
24 = s(2n) = 2na = 2*12,符合题意.
此时,s(3n) = 3na = 3*12 = 36.
q不为1时,s(n) = a[q^n-1]/(q-1),
12 = s(n) = a[q^n - 1]/(q-1),
24 = s(2n) = a[q^(2n)-1]/(q-1) = a[q^n-1]/(q-1) * [q^n + 1] = 12[q^n + 1],
2 = q^n + 1,q^n = 1.
此时,只能n为偶数,q=-1.
s(3n) = a[q^(3n)-1]/(q-1) = a[q^n-1]/(q-1)*[q^(2n) + q^n + 1] = 12[(1)^2 + 1 + 1]
= 12*3
= 36
综合,总有,
s(3n) = 36

s2n/sn=a1q^(2n-1)/(a1q^(n-1))=q^n=2,同理s3n/s2n=qn=2,故s3n=48.

等比数列{an}的前n项和为Sn、
那么Sn，S2n-Sn， S3n-S2n 。。。。同样是等比数列
Sn=12，S2n-Sn=12，那么S3n-S2n =12
所以S3n=24+12=36