作业帮已知等比数列{an}中,a4+a6=10,求a1a7+2a3a7+a3a9的值
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/29 23:41:20
已知等比数列{an}中,a4+a6=10,求a1a7+2a3a7+a3a9的值
已知等比数列{an}中,a4+a6=10,求a1a7+2a3a7+a3a9的值
已知等比数列{an}中,a4+a6=10,求a1a7+2a3a7+a3a9的值
a4+a6=10
a1q^3+a1q^5=10
a1(q^3+q^5)=10
a1q^3(1+q^2)=10
1+q^2=10/a1q^3
a1a7+2a3a7+a3a9
=a1a1q^6+a1a1q^2q^6+a1a1q^2q^6+a1a1q^2q^8
=a1²q^6+a1²q^8+a1²q^8+a1²q^10
=a1²q^(1+q^2)+a1²q^8(1+q^2)
=a1²q^6×10/a1q^3+a1²q^8×10/a1q^3
=10×a1(q^3+q^5)
=10×10
=100
设公比为q,则 a4+a6=a1*q^3+a1*q^5=10
所以 a1a7+2a3a7+a3a9=a1^2*q^6+2a1^2*q^8+a1^2*q^10
=(a1*q^3)^2+2a1*q^3*a1*q^5+(a1*q^5)^2
...
全部展开
设公比为q,则 a4+a6=a1*q^3+a1*q^5=10
所以 a1a7+2a3a7+a3a9=a1^2*q^6+2a1^2*q^8+a1^2*q^10
=(a1*q^3)^2+2a1*q^3*a1*q^5+(a1*q^5)^2
=(a1*q^3+a1*q^5)^2
=10^2
=100
a^b是a的b次方
收起
a1a7+2a3a7+a3a9
＝a4*a4+2a4a6+a6a6
=(a4+a6)²
=10²
=100