您好,设公比为r,则有:
a2 = ar
a3 = ar^2
又因为a1为a2和a3的等差中项,所以有:
a1 = (a2 + a3) / 2
a1 = (ar + ar^2) / 2
将a1代入a2和a3的表达式中,得到:
a2 = ar = a1 / r
a3 = ar^2 = a1 / r^2
因此,等比数列的通项公式为:
an = a1 * r^(n-1)
其中a1为等差中项,r为公比,n为项数。
an为公比不为1的等比数列a1为a2和a3的等差中项
∵{an}为等比数列,公比为q ∴an=a1*q^(n-1) ∵a2,(1/2)a3,a1为等差数列 ∴a3=a2+a1 即a1q²=a1*q+a1 ∴q²=q+1 q²-q-1=0 解得q=(1±√5)/
2 ∵an>0,∴q>
0 ∴q=(1+√5)/
2 ∴(a3+a4)/(a4+a5) =(a1q²+a1q³)/(a1q³+a1q⁴) =a1q²(1+q)/[a1q³(1+q)] =1/q =2/(1+√5) =(√5-1)/2