A geometric sequence is a sequence in which each pair of terms shares a common ratio. Another way of saying this is that each term can be found by multiplying the previous term by a certain number.

For example, in the sequence below, the common ratio is 2, because each term is 2 times the term before it.

1, 2, 4, 8, 16, ...

So if you know two terms in succession, it's easy to find the common ratio; simply divide the latter term by the previous one.

For example, if I know that the 10^th term of a geometric sequence is 24, and the 9^th term of the sequence is 6, I can find the common ratio by dividing the 10^th term by the 9^th term: 24 / 6 = 4. 

In the previous sequence, now that I know the common ratio is 4, I can easily find the 11^th term by multiplying the 10^th term by 4: 24 x 4 = 96.

Is there an equation that helps us find the n^th term?

Of course! Here it is:

a_n = a₁r^n-1

Does that make sense? Sure! You start with the first term, and multiply it by the common ratio n-1 times!

Once you have all this information under your belt, you can use it to solve all sorts of problems.

Example One: Find the fifth term of a geometric sequence if the second term is 12 and the third term is 18.

Solution: The common ratio is 18/12 or 3/2. Thus the fourth term is 27, and the fifth term must be 81/2.

Example Two: Find the second term and the common ratio if the third term is 4 and the fifth term is 16.

Solution: Don't get tricked by this one - it's tempting to say that the ratio must be 2, because 4 x 2 x 2 = 16, but in reality, the equation you're solving is r² = 4, and that has two solutions: 2 and -2! These lead to 2 and -2 as the possible values for the second term.

1.

If the fifth term of a geometric sequence is 16 and the sixth term is 32, what is the common ratio?

2.

If the second term of a geometric sequence is 81, and the fourth term is 9, what is the common ratio?

3.

Find the first term and ratio in a geometric sequence if the fifth term is 1, and the sixth term is -1.

4.

If the first term of a geometric sequence is 8 and the common ratio is 3/2, what is the fourth term?

5.

The first term of a geometric sequence is 5, and the fourth term is 40. What is the sixth term?

6.

The first term of a geometric sequence is one less than the sequence's ratio. If the second term is 30, what are the possible values of the third term?

7.

The second term of a geometric sequence is 5 more than the first term, and the third term is 10 more than the second one. What is the common ratio?

8.

The second term of a geometric sequence is 6 more than the first term, and the third term is 21 more than the first term. What is the common ratio?

9.

The first two terms of a geometric sequence add up to the third term. What is the common ratio?

10.

The first three terms of a geometric sequence are 18, 12, and 8. What is the first term in this sequence which is less than one? Write your answer as a fraction.