The number pattern that you see on your screen is known as an arithmetic sequence. An arithmetic sequence is any kind of pattern where the difference between any two numbers, or any two consecutive numbers I should say, is the same. For example, the difference between 23 and 25 is equal to 2. Likewise, the difference between 13 and 15 is also. equal to 2. Now we typically use the following formula when dealing with an arithmetic sequence and it's extremely important that you know what each part means. A is always going to be term number 1. N over here is the position number or we'll actually just say position.
Now it's extremely important that you know what we mean by position. For example we would say that the number 9 is at position 1. Whereas the 19 is at position 6. D is the difference between the numbers. And in this case, we said that the difference is 2. And then lastly, Tn is the value. We'll actually just call it the value. So for example, 19 is at position 6, but it has a value of 19. The number 25 is at position...
9, but it has a value of 25. Now let's see how this formula actually works. So let's say we wanted to work out this term over here. Let's pretend we could not see it, okay? So that we would be looking for a specific term. So we would just say t is equal to, oh, this n over here, it should actually be written like this, tn.
So 21 is at position number 1, 2, 3, 4, 5, 6, 7. So we could say t7, sorry, let me write that better, t7 is equal to a, a is always the first term, so that's 9, plus n, well that's at position 7, so we'll say 7 minus 1, times by d, which is 2. And if you had to go work that out, you would get 9 plus 12, which is 21. You see, so the formula helps us determine any particular number that we want. The formula could also be used in reverse. For example, let's say we know that we have a term 17, but we don't know what position it is at. So we use the formula and we fill in the t value.
Remember the t value stands for, well the t stands for the value. So the value is 17, but we don't know the position. I mean obviously we can see that the position is 5, but we're trying to see if the formula actually works. So 17 is equal to a, which is 9, plus n, we don't know, and then d is 2. So what we do is we take this 9 over to the left, so we end up with 8 equals to n minus 1 times 2. Now there's various ways of solving this, but I'm just going to divide the left-hand side by 2. So we're going to have 4 equals to n minus 1, and so n would be equal to 5. And so what that means is that the value of 17 takes place at position 5. Let's see if that is true. 1, 2, 3, 4. 5. So in summary, as soon as you have an arithmetic sequence, don't forget to use this formula, which will be given to you.