Question

A geometric sequence is

3
4
, 9, 108, 1296, 15,552, 186,624, ... . Which is the general term of the sequence?

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= ((3)/(4)) (12)^(n-1)` that is the nth term

Answer 2

The nth term formula for this geometric sequence is:

`a_(n)=(3)/(4)12^(n-1)`

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=(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=(108)/(9)=12`.

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

`a_(n)=(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)=(3)/(4)12^(1-1)=(3)/(4)\\a_(2)=(3)/(4)12^(2-1)=9\\a_(3)=(3)/(4)12^(3-1)=108\\a_(4)=(3)/(4)12^(4-1)=1296`

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