Question 117751
Notice that each term is increasing quickly. 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 we let n=0, we get


 
{{{a[0]=2*3^0=2*1=2}}} notice this is the first term



and if we let n=1, we get



{{{a[1]=2*3^1=2*3=6}}} notice this is the second term



and this generates the sequence 2, 6 , 18 , 54 , ....



So to find the 8th term, let n=7 (remember, we started at n=0)



{{{a[7]=2*3^7=2*2187=4374}}} Plug in n=7 and simplify




So the eighth term is 4,374