Answer:
7th term.
Step-by-step explanation:
The given series is
[tex]\log 3+\log 9+\log 27+\log 81+...[/tex]
We need to find the number of term log 2187.
Here, terms inside the log are 3, 9, 27, 81, ... are in G.P.
First term = 3
Common ratio [tex]=\dfrac{9}{3}=3[/tex]
nth term of a G.P. is
[tex]a_n=ar^{n-1}[/tex]
where, a is first term and r is common ratio.
[tex]2187=3(3)^{n-1}[/tex]
Divide both sides by 3.
[tex]729=(3)^{n-1}[/tex]
[tex]3^6=(3)^{n-1}[/tex]
On comparing both sides, we get
[tex]n-1=6[/tex]
[tex]n=7[/tex]
Since 7th term of the G.P. 3, 9, 27, 81, ... is 2187, therefore, 7th term of [tex]\log 3+\log 9+\log 27+\log 81+...[/tex] is log 2187.
