设{an}为等比数列,首项a1=4,公比q=3,求数列中前n项的所有不同两项乘积的和

Question

要过程，请速回答，谢谢！！

Answer 1

S=[a1an+a2an+……+a(n-1)an]+[a1a(n-1)+a2a(n-1)+……+a(n-2)a(n-1)]+……+a1a2
=an[a1+a2+……+a(n-1)]+a(n-1)[a1+a2+……a(n-2)]+……+a1a2
=a1*q^(n-1)*a1*[q^(n-1)-1]/(q-1)+a1*q^(n-2)*a1*[q^(n-2)-1]/(q-1)
+……+a1a2
=a1^2*[q^(2n-2)-q^(n-1)]/(q-1)+a1^2*[q^(2n-4)-q^(n-2)]/(q-1)+……+a1a2
=[a1^2/(q-1)][q^(2n-2)+q^(2n-4)+……+q^2-q^(n-1)-q^(n-2)-……-q^1]

q^(2n-2)+q^(2n-4)+……+q^2
=q^2*[(q^2)^n-1]/(q^2-1)
=(9/8)(9^n-1)
q^(n-1)+q^(n-2)+……+q^1
=q^1*(q^n-1)/(q-1)
=(3/2)(3^n-1)
所以S=(16/2)*[(9/8)(9^n-1)-(3/2)(3^n-1)]
=9(9^n-1)-12(3^n-1)
=9^(n+1)-12*3^n+3