各项都为正数的等比数列{an},前n项和为A,前n项的积为B,前n项的倒数为C求证：各项都为正数的等比数列{an},前n项和为A,前n项的积为B,前n项的倒数为C求证：B^2=(A/C)^n

Question

各项都为正数的等比数列{an},前n项和为A,前n项的积为B,前n项的倒数为C 求证：
各项都为正数的等比数列{an},前n项和为A,前n项的积为B,前n项的倒数为C 求证：B^2=(A/C)^n

Answer 1

设{an}公比为q A=a1+a2+...+an=a1+a1q+...+a1q^(n-1)=a1×[1+q+...+q^(n-1)] C=1/a1+1/a2+...+1/an=1/a1+1/a1q+...+1/a1q^(n-1)=1/a1q^(n-1)×[1+q+...+q^(n-1)] ∴A/C=a1/[1/a1q^(n-1)]=a1×a1q^(n-1)=a1×q^(n-1) ∴(A/C)^n=a1^2n×q^n(n-1) B=a1×a1q×...×a1q^(n-1)=a1^n×q^[1+...+(n-1)]=a1^n×q^[n(n-1)/2] ∴B^2=a1^2n×q^n(n-1) 即B^2=(A/C)^n