Deductive and inductive reasoning
Using inductive reasoning
What is the nth number in the following sequence? Let's see. The first term here is a 6. So over here I'll put the term. So this is the number. I'll just write num there. That's the number. And this is the term. So the first term here is a 6. It looks like we add 3 to that to get to a 9. Or maybe there's some other pattern here. So that's the second term, is 9. Then they go from 9 to 12. Looks like we add 3 again. So we keep adding 3. So the third term is 12, the fourth term is 15. And we need to think of a general expression that will give us the value of the nth term. The nth number in the following sequence. So let's think a little bit about it. We know that we're incrementing by 3 every time and we're starting at 6. So let's write these in terms of multiples of 3, since we know that that's the increment level. This first term right here is 3 times 2. The second term right here is 3 times 3. This third term is 3 times 4. And so it looks like it's 3 times one more than the actual term you are dealing with. Right? Here, if you take your term, you add 1 and then multiply that by 3, you get 6. Take your term, add 1, then multiply it by 3. Take your term, add 1, you get 4. Then multiply by 3. Take your term, add 1-- which would be 5-- and then multiply by 3, you get 15. So it looks like any time-- so if you have your nth term. So if you go all the way out here to your nth term, the value here would be you'd add 1 to it. So you'd have n plus 1, and then you'd multiply that by 3. So the nth term is 3 times n plus 1. Or if you wanted to distribute out the 3, it'd be 3n plus 3. And just make sure that works for all of them. The first term, 3 times 1 is 3 plus 3, that works. That's 6. The second term, 3 times 2 plus 3 is 9. So it works for all of these terms. So that is the nth term. We have to assume that this pattern continues.