等比数列An中,若Sn=12,S2n=24,求S3n
问题描述：
等比数列An中,若Sn=12,S2n=24,求S3n
答
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