GEOMETRIC PROGRESSION

[Guest]

Greetings. I am in need your assistance.
I am currently studying geometric progressions when I found an example. It is an example that is under the subtitle of geometric median of extreme terms of a G,P

and it says:

Given an geometric progression so that a1*a9=9*a3*a7, calculate the common ratio.

Then It says :Observe that a a1*a9=a5^2 and a3*a7=a6^2=9(a5q)^2 and, so q=1/3

But how is it that a3*a7=a6^2? and how did he get the ratio?
Could you please help me understand it?

ps :the numbers in a1,a9,a3 are to be seen as the index( to better clarify)
Your help is deeply appreciated.
Thank You.

[Guest]

The right thing is [tex]a_{3 }[/tex][tex]a_{7 }[/tex]=([tex]a_{5 })^{2}[/tex]

[tex]a_{3 }[/tex][tex]a_{7 }[/tex]=([tex]a_{1 }[/tex][tex]q^{2}[/tex])([tex]a_{1 }[/tex][tex]q^{6}[/tex])=[tex]a_{1 }^{2}[/tex].[tex]q^{8}[/tex]=[tex](a_{1 }q^{4})^{2}[/tex]=[tex]a_{5 }^{2}[/tex]

Example 1;2;4;8;16;32;64;...

[tex]a_{3 }[/tex][tex]a_{7 }[/tex]=4.64=256=[tex]16^{2}[/tex]=[tex]a_{5 }^{2}[/tex]

[Guest]

The right thing is [tex]a_{3 }[/tex][tex]a_{7 }[/tex]=([tex]a_{5 })^{2}[/tex]

[tex]a_{3 }[/tex][tex]a_{7 }[/tex]=([tex]a_{1 }[/tex][tex]q^{2}[/tex])([tex]a_{1 }[/tex][tex]q^{6}[/tex])=[tex]a_{1 }^{2}[/tex].[tex]q^{8}[/tex]=[tex](a_{1 }q^{4})^{2}[/tex]=[tex]a_{5 }^{2}[/tex]

Example 1;2;4;8;16;32;64;...

[tex]a_{3 }[/tex][tex]a_{7 }[/tex]=4.64=256=[tex]16^{2}[/tex]=[tex]a_{5 }^{2}[/tex]

Thank you!