Learn to Identify Patterns and Make Conjectures

TenMarks teaches you how to identify patterns and make conjectures.

Learn to Identify Patterns and Make Conjectures In this lesson, let’s learn how we identify patterns and make conjectures. We’ll do three problems but before we try the problems, let’s learn about what is a conjecture. Actually I’ll explain it as we look at the first problem. The first problem wants us to find the next term in the pattern given to us which is January, March, May, and July. If we look at what they have given to us, we see that January is the first month of the year, March is the third month, May is the fifth month, and July is the seventh month and so on. So the pattern that I'm seeing is we go to the first month, then we go to third month, fifth month, seventh month, which is all odd numbers right? So we’re seeing that all of these are odd numbers. So this is the first odd number, this is the second odd number, next odd number, next odd number and so on. So based on this, the next term should be that odd number after seven which is nine and the ninth of the year is September. So the next should be September. So notice what we did here. What we did is we looked at the pattern given to us and based on what we could figure out this process is called inductive reasoning. So I use examples to introduce what the pattern extended will be. So I'm going from January to the next alternate month, January, March, May, July, and September. So using inductive reasoning, I figured out that the rule here is it’s every alternate month right? So the pattern is every alternate month or odd months in a year. Because this rule is based on inductive reasoning, I don’t know for sure the rule has not been given to me. I look at the examples and based on that, I'm reasoning that this is the rule such a rule is called a conjecture. So a conjecture is certainly true for the example given to us but we are never 100% sure if the conjecture will hold good forever because it is based on just a few examples based on which we did the inductive reasoning. So, that’s what's inductive reasoning and a conjecture. Now, let’s learn how to use this in a couple of other problems. So when we look at completing a conjecture which says a rule in terms of N for the sum of the sum of the first N positive integers. So let’s use some examples. So one positive integer, first one positive integer when N=1. What is the first positive integer? It’s 1 right. What's the sum? It's 1. When N is 2, the first 2 positive integers are 1 and 3. Two odd positive integers are 1 and 3. What’s the sum? 1+3 is 4. When N is 3 which means I have to do the first 3 N odd positive integers, so first three odd positive integers that’s 1, 3 and 5. What's the sum of these, 5+3+1 is 9. Similarly, 4 is 1, 3, 5 and 7, 16. So this is the end result. What is this? 1², 2², 3², 4² etcetera. So based on this a conjecture looks like if N is 1 the sum of—if N = 1 the sum of N positive odd integers is N², right? So this term, in this rule that we were looking at can be N². That’s what we imagine it is. This is the conjecture. And the way we got that is by drawing a table with the examples and seeing the pattern. The pattern is 1², 2², 3², 4² and so on. So we have every reason to believe that this conjecture is indeed the rule but we don’t know that for a fact because I've only looked at four examples. So, we can say this is the conjecture. Let’s do one more problem. It says a laboratory culture contains 150 bacteria. After 20 minutes, the culture contains 300 bacteria. So I'm going to basically say N = 1. So the first time, let’s look at number of minutes is 0 and it has 150 bacteria, number of bacteria is 150. And on this two, in these 20 minutes, I have 300 bacteria. After one hour, one hour is 60 minutes right? 60 minutes means N is 4 or actually N is 1, 2, 3 would be 40 minutes, 4 would be 60 minutes. I have 1,200 bacteria. Make a conjecture about the rate which increases. So here’s what I'm given right, 0 minutes, 20 minutes, 40 minutes, 60 minutes. And 0 minutes it’s 150, 20 minutes is 300, 60 minutes is 1,200. So if I put 600 in the middle it went from 300 to 600 after N = 3. If N = 4 it’s 1,200. What is the rule that I can think about? This is 2 x 0 x 150. This is 2 x 1 x 150; 2² is 4 x 150 or 23 x 150. So it looks like the pattern is if instead of 1, 2, 3, 4 I put 0, 1, 2, and 3. That’s the pattern right? Zero, 0, 150; when N = 1, it’s 2¹ x 150. If N = 2, 2² x 150 and so on. So it looks like that rule is the number of bacteria = 2N x 150, where N = time in 20 minutes increments. So, if N = 3 which is 60 minutes, it is 2 x N. If N = 2. So N = time in 20 + minutes increments. So if we have 20 minutes, 40 minutes, 60 minutes. If N =3 which is 60 minutes it is 2³ x 150. Again, what we did was we used the data given to us to come up with examples and based on that came up with a pattern and a rule that makes sense.