用S表示等差数列{an}的前k项和求证：Sk、S2k-Sk、S3k-S2k也成等比数列.

题目

用S表示等差数列{an}的前k项和求证：Sk、S2k-Sk、S3k-S2k也成等比数列.

答案

用S表示等比数列{an}的前k项和求证：Sk、S2k-Sk、S3k-S2k也成等比数列.
Sk=a1(1-q^k)/(1-q)=[a1/(1-q)]*(1-q^k)
S2k=a1(1-q^2k)/(1-q) S2k-Sk=[a1/(1-q)]*(q^k-q^2k)
S3k=a1(1-q^3k)/(1-q) S3k-S2k=[a1/(1-q)]*(q^2k-q^3k)
(S2k-Sk)/Sk=(q^k-q^2k)/(1-q^k)=q^k
(S3k-S2k)/(S2k-Sk)=(q^2k-q^3k)/(q^k-q^2k)=q^k
Sk、S2k-Sk、S3k-S2k也成等比数列.