等比数列{an}中的各项均为正数,且a5a6+a4a7=54,则log3a1+log3a2+...+log3a10=

题目

等比数列{an}中的各项均为正数,且a5a6+a4a7=54,则log3a1+log3a2+...+log3a10=

答案

由于{an}为等比数列
则:a5a6=a4a7=a3a8=a2a9=a1a10
又a5a6+a4a7=54
则:
2a5a6=54
a5a6=27
则:
log3(a1)+log3(a2)+...+log3(a9)+log3(a10)
=log3[a1*a2*a3*...*a10]
=log3[(a1a10)*(a2a9)*...*(a5a6)]
=log3[27*27*...*27]
=log3[27^5]
=log3[3^15]
=15log3[3]
=15
祝你学习愉快