Learn about Non-Arithmetic Sequences

Now let's learn about non-arithmetic sequences. The problem says for us to identify the pattern in the following sequence and then find the missing terms. The sequence given to us is 45, 46, 43, 44, 41, blank and blank. We have to find the blank and blank.

First, let's understand what is a sequence. A sequence is an ordered set of numbers. It's an ordered set of numbers that follows a pattern and each number is called a term. Each number is called a term. So, the first term is 45. The second term is 46. The first term is 45, the second term is 46. The third term is 43 and so on.

So let me teach you how to do this. Obviously, this is not an arithmetic sequence because the difference between each one of these is not the same. The difference between 45 and 46 is one, 43 and 46 is (-3) et cetera, so it's not the same. But let's determine what the pattern is. So I'm just going to write down the terms; 45, 46, 43, 44, 41 et cetera. In order to do this, the best thing to do is to look at the first three terms and try and determine the pattern.

So, how do I get from 45 to 46? Well, I add one. To get from 46 to 43, I subtract three. Now let's look at how do I get from 43 to 44. Well, I again add one. 44 to 41, I subtract three. As you can see, I can start to see a pattern. I add one, then to get the next term, I subtract three then I add one, then I subtract three. So what do we know? Well, the pattern says that the second term equals the first term plus one and the third term equals the second term minus three. I'll take a little bit more space.

So as we can see, the fourth term equals the third term plus one and so on. I can generalize this pattern and say that every even numbered term, second and fourth, every even numbered term equals the previous term plus one and every odd term, odd numbered term equals the previous term minus three. As we can see, the second term which is even number; we can calculate by taking the previous term and adding one. Third term which is odd, I take the previous one and subtract three.

Similarly the fifth term, I take the previous term and subtract three. So now that we know the pattern, we can determine the missing terms. So what do we have to do? One, two, three, four, five; we have to find the sixth and the seventh term. So the sixth term should be, well if it's an even numbered term it should be the previous term which is the fifth term plus one. What is the fifth term? One, two, three, four, five, fifth term is 41 + 1. So the sixth term is 42.

The seventh term is an odd term; so an odd numbered term is the previous term minus three. So the seventh term would be the sixth term minus three. The sixth term we just knew is 42, so it's 42 – 3 which is 39.

So, the key thing that we've remembered is when we have a sequence, we need to look at the numbers in the sequence and try and establish the pattern. As we could see, we saw a plus one, minus three, plus one, minus three, et cetera which meant that the second term was the first term plus one. The fourth term was the third term plus one. So every even term was the previous term plus one. But the third term is the second term minus three.

Similarly, the fifth term was the fourth term minus three. So every odd numbered term was the previous term minus three. Once we have the pattern, then it was easy for us to determine the sixth and seventh term. Since sixth was an even term, it would be the previous term which is the fifth one plus one, 41 + 1, it needs to be 42 and the seventh tern is the sixth term minus three which is 42 – 3 which is 39. So the final answer is 42 and 39.