Watch this video lesson to learn the formula for the general term of a geometric sequence. You will learn how to write your own general term given a particular geometric sequence as well as the reason for doing this.
A Geometric Sequence
In this video lesson, we are going to be talking about geometric sequences and how to use the formula for the general term of a geometric sequence. We learn about this because we encounter geometric sequences in real life, and sometimes we need a formula to help us find a particular number in our sequence. We define our geometric sequence as a series of numbers, where each number is the previous number multiplied by a certain constant. This constant varies for each series but remains constant within one series.
Wondering how this kind of series relates to the real world? You might be surprised to learn that this kind of thing happens all the time, and you might even be involved in it yourself. Imagine that you wanted to spread the message that Friday is the new red scarf day, and everyone should wear a red scarf. First, you send your message to five of your friends. So, we started off with one person, and now we have five new people.
In the message, you ask your friends to send this message about the new red scarf day to five of their friends to help you spread the word. You also ask them to keep the ball rolling and ask their friends to each spread the word to five of their friends. Now we have 25 new people who know about the message. If each of these 25 people tells 5 of their friends, how many new people will know about the red scarf day? One hundred twenty-five people. So we have a geometric sequence of 1, 5, 25, 125... because each number is the previous multiplied by a constant, the common ratio. The common ratio in our case is five. Each of our numbers is multiplied by five to get to the next.
The General Term
We actually have a formula that we can use to help us calculate the general term, or nth term, of any geometric sequence. It is x sub n equals a times r to the n - 1 power.
In this formula, x sub n stands for the particular number in that sequence. So x sub 4 stands for the fourth term in our sequence. The n stands for the term that we are looking at. If n is 10, then we are looking for the tenth term in our sequence. The r stands for the common ratio, the multiplication constant that is used in the geometric sequence to calculate each successive number or term. We plug in our a and r for our geometric sequence when we want to use this formula.
Why We Use It
Why do we use this formula when we can just write our sequence to find that particular term that we want? Well, this works well for terms that are in the beginning of our series. We can easily find our third, fourth, and fifth terms by just writing out our sequence. For our red scarf example, we already have the first four terms.
But what if we want to find out how many people will be reached with the red scarf message on the 50th round? What then? Do you want to sit there and calculate the numbers until you reach the 50th term? Or would you rather use the formula for the general term and calculate it? I would rather use the formula. And this is exactly why we have this formula. It makes it easy to calculate any number in our sequence.
How to Use It
Let's see how we can use it. For our red scarf, our r is 5 and our a is 1 since our beginning term is 1. Plugging these into our general term, we get x sub n equals 1 times 5 to the n - 1 power.
This becomes the equation for the general term of our red scarf geometric sequence. Because my first term is 1 and anything multiplied by 1 is itself, I have rewritten my equation without the times 1 part. If my first term was anything but 1, I would keep it there.
To find a particular number in our sequence, we simply plug in the n value associated with that number to find our number. So for the 50th number, my n will be 50. Plugging in 50 for n, I get x sub 50 = 5 to the 50 - 1 power, or 5 to the 49th power. 5 to the 49th power is 1.77 * 10^34. My 50th number in the sequence is then 1.77 * 10^34, a very large number. That's a lot of new people after only 50 rounds.
Let's review what we've learned now. We've learned that a geometric sequence is a series of numbers where each number is the previous number multiplied by a certain constant. This multiplication constant is also referred to as the common ratio. The general term, or nth term, of any geometric sequence is given by the formula x sub n equals a times r to the n - 1 power, where a is the first term of the sequence and r is the common ratio.
We use this formula because it is not always feasible to write out the sequence until we reach our desired number. To use this formula, we plug in a and our r for our particular geometric sequence and then we plug in our desired n to find our desired number in the sequence.
You should have the ability to do the following after watching this video lesson:
- Define geometric sequence and common ratio
- Identify the formula for the general term
- Explain the benefit of using this formula