Question 179479
First, let's find the formula (or rule) that ties all of these terms together:



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


First divide the 2nd term 3 by the 1st term 1 to get  

{{{3/1=3}}} 

 
Now divide the 3rd term 9 by the 2nd term 3 to get  

{{{9/3=3}}} 

 
Now divide the 4th term 27 by the 3rd term 9 to get  

{{{27/9=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}}}. What this tells us is that to get the next term, simply multiply the current term by 3.



Now since we've started at 1, this means that {{{a=1}}}


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


{{{a[n]=1*3^n}}} or simply {{{a[n]=3^n}}} where {{{n}}} starts at {{{n=0}}}



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Now let's find the 9th term. 



The 9th term will occur when {{{n=8}}} (since we started at {{{n=0}}})



{{{a[n]=3^n}}} Start with the given sequence



{{{a[8]=3^8}}} Plug in {{{n=8}}}



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



So the ninth term is {{{a[8]=6561}}}