Question 119215

Notice that each term is increasing exponentially. So this sequence might be a geometric sequence. To find out, let's simply divide the terms.


First divide the 2nd term 6 by the 1st term 2 to get  

{{{6/2=3}}} 

 
Now divide the 3rd term 18 by the 2nd term 6 to get  

{{{18/6=3}}} 

 
Now divide the 4th term 54 by the 3rd term 18 to get  

{{{54/18=3}}} 

 

So if we pick any term and divide it by the previous term, we'll always get 3. This is the common ratio between the terms. So this means that {{{r=3}}}.


Now since we've started at 2, this means that {{{a=2}}}


Since the general geometric sequence is {{{a[n]=ar^n}}}, this means the sequence is


{{{a[n]=2*3^n}}}



Notice if n=0, then 



{{{a[0]=2*3^0=2*1=2}}}


if n=1, then 



{{{a[1]=2*3^1=2*3=6}}}


if n=2, then 



{{{a[0]=2*3^2=2*9=18}}}



etc...





{{{a[8]=2*3^8}}} Now to find the 9th term, plug in n=8 (since we started at zero n=8 is the 9th term)




{{{a[8]=2*6561}}} Raise 3 to the 8th power to 6,561



{{{a[8]=13122}}} Multiply 2 and 6,561 to get 13,122




So the 9th term is  13,122