Question

A geometric sequence is

Answer 1

An=a1(r)^(n-1)
r=common ratio
a1=first term
given
a1=3/4 or 0.75

th ecommon ration is a term divided by previous term
108/9=12
r=12

$a_n= (\frac{3}{4}) (12)^{n-1}$ that is the nth term

Answer 2

Answer:

The nth term formula for this geometric sequence is:

$a_{n}=\frac{3}{4}12^{n-1}$

Step-by-step explanation:

To find the general term of the sequence or the nth term of geometric sequence we can use the formula:

                            $a_{n}=a_{1}r^{n-1}$

where

r = common ratio

a1 = first term of the sequence

$a_{n-1}$ = 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 $r=\frac{a_{n}}{a_{n-1}}$. In the geometric sequence given we can choose for example 108 as $a_{n}$ and 9 as $a_{n-1}$. Applying this equation we have $r=\frac{108}{9}=12$.

Using the nth term formula for a geometric sequence we have:

$a_{n}=\frac{3}{4}12^{n-1}$

To be sure that this formula works we can replace some values of n to find the elements in the sequence for example:

$a_{1}=\frac{3}{4}12^{1-1}=\frac{3}{4}\\a_{2}=\frac{3}{4}12^{2-1}=9\\a_{3}=\frac{3}{4}12^{3-1}=108\\a_{4}=\frac{3}{4}12^{4-1}=1296$

And these results are in the sequence given, so we prove that the formula works.