Option B:
The 12th term is 354294.
Solution:
Given data:
[tex]a_4=54[/tex] and [tex]a_7=1458[/tex]
To find [tex]a_{12}:[/tex]
The given sequence is a geometric sequence.
The general term of the geometric sequence is [tex]a_n=a_1\ r^{n-1}[/tex].
If we have 2 terms of a geometric sequence [tex]a_n[/tex] and [tex]a_k[/tex] (n > K),
then we can write the general term as [tex]a_n=a_k\ r^{n-k}[/tex].
Here we have [tex]a_4=54[/tex] and [tex]a_7=1458[/tex].
So, n = 7 and k = 4 ( 7 > 4)
[tex]a_7=a_4\ .\ r^{7-4}[/tex]
[tex]1458=54\ . \ r^3[/tex]
This can be written as
[tex]$r^3=\frac{1458}{54}[/tex]
[tex]$r^3=27[/tex]
[tex]$r^3=3^3[/tex]
Taking cube root on both sides of the equation, we get
r = 3
[tex]a_{12}=a_7\ .\ r^{12-7}[/tex]
[tex]=1458\ .\ r^5[/tex]
[tex]=1458\ .\ 3^5[/tex]
[tex]a_{12}=354294[/tex]
Hence the 12th term of the geometric sequence is 354294.
