Answer:
The nth term formula for this geometric sequence is:
[tex]a_{n}=\frac{3}{4}12^{n-1}[/tex]
Step-by-step explanation:
To find the general term of the sequence or the nth term of geometric sequence we can use the formula:
[tex]a_{n}=a_{1}r^{n-1}[/tex]
where
r = common ratio
a1 = first term of the sequence
[tex]a_{n-1}[/tex] = the term before the nth term
n = number of terms
In a Geometric Sequence, each term is found by multiplying the previous term by a constant. This constant is the common ratio and the way to find it is [tex]r=\frac{a_{n}}{a_{n-1}}[/tex]. In the geometric sequence given we can choose for example 108 as [tex]a_{n}[/tex] and 9 as [tex]a_{n-1}[/tex]. Applying this equation we have [tex]r=\frac{108}{9}=12[/tex].
Using the nth term formula for a geometric sequence we have:
[tex]a_{n}=\frac{3}{4}12^{n-1}[/tex]
To be sure that this formula works we can replace some values of n to find the elements in the sequence for example:
[tex]a_{1}=\frac{3}{4}12^{1-1}=\frac{3}{4}\\a_{2}=\frac{3}{4}12^{2-1}=9\\a_{3}=\frac{3}{4}12^{3-1}=108\\a_{4}=\frac{3}{4}12^{4-1}=1296[/tex]
And these results are in the sequence given, so we prove that the formula works.
