Deductive and inductive reasoning
Jill looked at the following sequence. 0, 3, 8, 15, 24, 35. And it just keeps going, I guess, with a dot, dot, dot. She saw that the numbers were each 1 less than a square number. 0 is 1 less than 1, which is a square number. 3 is 1 less than 4. 8 is 1 less than 9. 15 is 1 less than 16. Yeah, they were all 1 less than a square number. And conjectured that the nth number would be n squared minus 1. Now conjecture, that sounds like a very fancy word. When someone makes a conjecture, they conjecture, that just means that they're making a statement that seems, or they're making a proposition that seems likely to be true. It seems like a very reasonable thing to say. But it's not definitely true. So she conjectured that the nth number would be n squared minus 1. The reason why this is a conjecture as opposed to a 100% definitely true statement, is we don't know whether this pattern continues. She's just going off of the pattern that she saw so far and she just generalized it. She just assumes that it keeps on going. But we don't know whether it necessarily keeps going. Maybe the next number, you would expect it to be 48, but maybe it's not 48. Maybe it's something weird. Maybe it's 2. Maybe it's 500. And so the conjecture wouldn't hold up if you were to see that, but based on the evidence you see so far it seems completely reasonable that this pattern would continue. And so she conjectured that the nth number would be n squared minus 1. Completely reasonable. Now did Jill use inductive reasoning? Yes, she used inductive reasoning. That's what inductive reasoning is. You see a pattern. In this case, every term in this sequence so far was-- if it's the third term, it was 3 squared minus 1. The fourth term is 4 squared minus 1. The fifth term is 5 squared minus 1. So she saw the pattern and she just generalized it to say, well, I think or I've conjectured that the nth number will be n squared minus 1. That's what inductive reasoning is all about. You're not always going to be 100%, or you definitely won't be 100% sure that you're right, that the nth number will be n squared minus 1. But based on the pattern you've seen so far, it's a completely reasonable thing to-- I guess you could say-- to induce.