Chapter 6: Rationals: Multiplication, Division, and Applications
Study the sequence: 1, 2, 4, 8, 16, ... What can you say about this sequence? We could say that the first term is 2^0, the second 2^1, and generally, the nth term in the sequence 2^(n-1). This is a very correct way to describe this sequence. However, what we are going to be discussing today is much simpler...
In the above sequence, divide the second term by the first term. (A term is simply a number in the sequence.) Did you get 2? Now try dividing the fourth term by the third term. 2 again? In fact, if you divide any term in the above sequence by the term to its immediate left, you will get an answer of 2. A sequence that has this characteristic is called a geometric sequence. In this case, 2 is the common ratio of the sequence. Consequently, you can multiply any term in the sequence by the common ratio to obtain the next term of the sequence. These definitions are important for a formula that will be introduced in a short while...
Study the following sequences and determine whether or not each sequence is geometric. If it is not, figure out why.
1. 4, 8, 12, 16, 20, ...
2. 2.5, 5, 10, 20, 40, ...
3. 5, -15, 45, -135, 405, ...
4. 4, 2, 1, 0.5, 0.25, ...
5. 2, 2, 4, 16, 65536, ...
Solutions to examples:
1. This sequence is not geometric because there is not a common ratio. This type of sequence is called an arithmetic sequence. If you subtract any term from the term to its immediate right, you obtain the same number (in this case, 4) which is called the common difference of the sequence.
2. This sequence is a geometric sequence with a common ratio of 2.
3. This sequence is a geometric sequence with a common ratio of -3.
4. This sequence is a geometric sequence with a common ratio of 0.5.
5. This sequence is not geometric because there is not a common ratio. It works in sets of three terms. Look at the first three terms. The second term is called the base, the first term is called the exponent, and the third term is the result. E.g. 2^2 = 4
Sometimes you may want to know a certain term of a sequence. If the desired term is one that's early in the sequence, then it should not be too hard to find. However, what if you needed to find the 100th term of a sequence. You really do not want to sit and multiply by two 100 times, do you?!? Luckily, we have a formula that will easily calculate any term of a geometric sequence that we want (as long as the numbers do not get too big!!). Here it is:
a*(r)^n-1, where a is the first term, r is the common ratio, and n is the number of term that you want to find.
Let's look at example 2 from above. Let's say that we want to find the 10th term of this sequence. Then n = 10. We see that 2.5 is our first term so a = 2.5. Finally, as you have hopefully computed, the common ratio of this sequence is 2 (r = 2). So, after plugging these numbers into our formula, we find that the 10th term is 2.5*(2)^9 = 1280. Since 10 is not such a large number we can check our formula for accuracy. The last term above is 40, so 40*2 = 80 (#6)
80*2 = 160 (#7)
160*2 = 320 (#8)
320*2 = 640 (#9)
640*2 = 1280 (#10); therefore, our formula works!
Click here for an interactive demonstration of geometric sequences.