Question

若等比数列{an}的各项均为正数，且a5a6=9，则log3a1+log3a2+......+log3a10=？（要过程） 谢谢了！！！

Answer 1

设q为公比，a5*a6=a5*a5*q=a5^2 * q=(a1*q^4)^2 *q=a1^2 * q^9=9
log3a1+log3a2+......+log3a10=log3a1+log3a1*q+......+log3a1*q^9
=log3 [(a1*a1...*q1) * (q*q^2*...q^9)]
=log3 (a1^10*q*q^2*...q^9)
=log3 (a1^10 * q^45)
=5*log3(a1^2 * q^9) (已知a1^2 * q^9=9)
=5*log3(9)
=15

Answer 2

因为a5a6=9--->a1^2q^9=9
又a1a10=a1*a1q^9=9,a2a9=a1q*a1q^8=a1^2*q^9=9,a3a8=a1^2*q^9=9,
a4a7=a1^2*q^9=9
所以a1a2a2……a9=(a5a6)^5=9^5=3^10
log(3)a1+log(3)a2+……+log(3)a9
=log(3)(a1a2……a9)
=log(3)3^10
=10.

Answer 3

我们老师刚刚讲过这道题

因为a5a6=9--->a1^2q^9=9
又a1a10=a1*a1q^9=9,a2a9=a1q*a1q^8=a1^2*q^9=9,a3a8=a1^2*q^9=9,
a4a7=a1^2*q^9=9
所以a1a2a2……a9=(a5a6)^5=9^5=3^10
log(3)a1+log(3)a2+……+log(3)a9
=log(3)(a1a2……a9)
=log(3)3^10
=10.

Answer 4

log3a1+log3a2+…+log3a10=log3（a1a10）+log3（a2a9）+…log3（a5a6）=5log3（a5a6）=10

Answer 5

因为a5a6=9--->a1^2q^9=9
又a1a10=a1*a1q^9=9,a2a9=a1q*a1q^8=a1^2*q^9=9,a3a8=a1^2*q^9=9,
a4a7=a1^2*q^9=9
所以a1a2a2……a9=(a5a6)^5=9^5=3^10
log(3)a1+log(3)a2+……+log(3)a9
=log(3)(a1a2……a9)
=log(3)3^10
=10.