Find the 9th term of the following geometric sequence: 1, –3, 9, –27.

When you are given a geometric sequence your first thought should be, what is the pattern? It can be simple, and sometimes more complex. A pattern, is simply something that is repeated to yield results. 

Most of the time you can't just find the answer. You have to think it through and test several different patterns. Common patterns include the previous number multiplied by some number, "n." Another common pattern is the addition or subtraction of some amount, "n." 

In this case of our geometric sequence: 
1, -3, 9, -27.

We can test a few different patterns. If the pattern is the correct one then if it works on one of them then it will work on all of them. Now we just need to test a few different patterns. 

1n = -3

The above equation is me testing the multiplication pattern. So, we just need to solve for n by dividing 1 on both sides yielding the results: (-3/1 = -3)

n = -3

Let's also try the addition/subtraction pattern
1 + n = -3

(Keep in mind that to test either one of these (addition/subtraction) the equation above will be sufficient, because if you solve for "n" you will get a negative number if it was a subtraction problem) 

Solve for n by subtracting 1 on both sides. 

n = -4

Keep in mind the numbers 1 came from the first number in the sequence. While you can choose any of the numbers in the sequence given to start with (besides -27), the amount you make it equal to has to be the number to the right of the chosen number. 

Now we need to test it with a few of the other numbers in the pattern, such as -3. This time plug in the values we got for n in the previous problem. 

Multiplication Test:
-3n = 9
n = -3

-3*-3 = 9

That one checks out. 

Addition/Subtraction Test: 
-3 + n = 9
n = -4

-3 - 4 = 9
-7 = 9

That is NOT true. So that pattern fails. 

If you keep checking the times -3 pattern, it works out. So now we need to the 9th term. We currently have 4 terms, so we need 5 more. To get to them we simply keep repeating the pattern over and over again. 

-27 * -3 = 81 (5th)
81 * -3 = -243 (6th)
-243 * -3 = 729 (7th)
729 * -3 = -2187 (8th)
-2187 * -3 = 6561 (9th)

Our answer is, 6561!