已知等比数列{an},公比为q,|q|不等于1,比较大小:(a3)^2+(a7)^2和(a4)^2+(a6)^2

问题描述:

已知等比数列{an},公比为q,|q|不等于1,比较大小:(a3)^2+(a7)^2和(a4)^2+(a6)^2
如何比较阿?

M=a3^2+a7^2=(aq^2)^2+(aq^6)^2=a^2q^4(1+q^8)N=a4^2+a6^2=(aq^3)^2+(aq^5)^2=a^2q^4(q^2+q^6)(M-N)/(a^2q^4)=1+q^8-(q^2+q^6)=q^2(q^6-1)-(q^6-1) =(q^2-1)(q^6-1)=(q^2-1)(q^2-1)(q^4+q^2+1) ...