已知{an}是等比数列1）a5^2=a3·a7是否成立?a5^2=a1·a9是否成立?2)an^2=a(n-2)·a(n+2)[n>2]是否成立?an^2=a(n-k)·a(n+k)[n>k>0]是否成立?注意：括号中均为角码“（）”

Question

已知{an}是等比数列
1）a5^2=a3·a7是否成立?a5^2=a1·a9是否成立?
2)an^2=a(n-2)·a(n+2)[n>2]是否成立?an^2=a(n-k)·a(n+k)[n>k>0]是否成立?
注意：括号中均为角码“（ ）”

Answer 1

1 a5=a1q^4
a3=a1q^2,a7=a1q^6
所以a5^2=a1^2*q^8=a3*a7
a9=a1q^8
所以a5^2=a1*a9
2 an=a1q^(n-1),a(n-2)=a1q^(n-3),a(n+2)=a1q^(n+1)
所以an^2=a1^2*q^(2n-2)=a(n-2)·a(n+2)
a(n-k)=a1q^(n-k-1),a(n+k)=a1q^(n+k-1)
a(n-k)a(n+k)=a1^2*q(2n-2)=an^2