Hey everybody, this is Paul. In this tutorial I'm going to be doing a proof by mathematical induction. So I'm going to be proving that 1 plus 2 plus 3 plus all the way up to n is equal to n times n plus 1 divided by 2. So what we want to do first is we want to check the base case.
So this will be the basis step, and we're checking the base case. We're checking the most simple basic case that we can come up with here. And it's going to be when n... equals 1. And we're just going to check that for n equals 1, the left-hand side and the right-hand side is true.
Well, if we're looking at the n equals 1 term on the left-hand side, then it's just basically the first term. So the first term happens to have the value 1. So we're going to write down the value 1 here to check the left-hand side. And then the right-hand side When n equals 1, we just simply replace the n's with the value 1, so this becomes 1 times 1 plus 1 divided by 2. And then we just are simply checking to see if the right-hand side is equal to the left-hand side for this base case.
1 plus 1 is 2, we multiply that by the 1 on the left here, divide that by the 2 on the bottom. 1 times 2 is 2, divided by the 2 on the bottom is simply equal to 1. So for the base case for n equals 1, we have basically... 1 on the left hand side equals 1 on the right hand side.
So I think that's simple enough. We can just erase it. So basically for the basis case we have 1 equals 1 and that is a true statement. And so since the basis case is true we can proceed to the next step. So the next step is going to be the induction step.
And basically what we're going to do with the induction step is we're going to do a couple things. We're basically just going to assume that this statement is true for some value n equals k. So we're going to assume that it's true for n equals k.
And so basically let's go ahead and do that now. Let's just say that we have 1 plus, let's write it over here to give myself some more room. So basically for this step we have 1 plus 2 plus 3. plus all the way up to some value k, we're saying that we're just going to assume that that's equal to the right hand side. So plugging in k where n's are, k plus 1 divided by 2. So we've assumed that this is true.
And so then the next thing we want to do is we want to show that it is true for n is equal to k plus 1. So we've assumed this step is true for n equals k. We're going to show that it's true for n equals k plus 1, and we're going to do that based off of this assumption. So for n equals k plus 1, we basically have 1 plus 2 plus 3 plus all the way up to k, and then we go one more further to k plus 1, and then we're going to also plug in k plus 1 where the n's are here. So this is going to be k plus 1 for that part. And that's multiplied by k plus 1, plugging that in where the n is.
And then we have plus 1. We could put this in parentheses to kind of set it apart, but really we're just going to be adding these anyway, so I'm just going to leave it like this. And we're going to divide that by the 2 here. So we're assuming this is true, and we're going to show that this is true based off of this assumption.
So what we're going to do with this assumption is if you notice right here we have 1 all the way up to k. So that's the same as what we have right here. And since we're assuming this is true, then we're basically saying that this right here we can use to replace this.
Because these are the same. So we're just going to replace all this stuff with the right hand side of this assumption. So this becomes k times k plus 1. divided by 2, and then we add that to the k plus 1 over here.
And then we're just going to check, is that equal to k plus 1? And then here, this is really k plus 1 plus 1, or k plus 2, divided by the 2. And then one more thing I want to do is notice how this is divided by 2, this is divided by 2. I'd like to make a denominator of 2 as well to match for this term, so I'm simply just going to multiply this by the number 1, and I'm going to choose 2. divided by 2 to be my number 1. Since 2 divided by 2 is equal to 1, I just multiplied this by 1. Anything times 1 doesn't change its value. So now I have all three of these terms with the same common denominator. So basically, if we can show that this statement is true now, then we've basically completed the mathematical induction proof.
So If you notice, all of these are 2 now. We don't even really need to look at the denominators. They're all the same. So now if we can just show that the numerators are the same, then we've basically completed the proof. So let's just rewrite the numerators to make this a little bit more simple.
k times k plus 1 plus 2 times k plus 1 is equal to k plus 1 times k plus 2. So now we're just going to... Just make sure that the numerators are the same, since we already know the denominators are the same. So k times k is k squared. Then we do the plus, distributing the k to the second term. k times 1 is k.
And then we're going to add that to 2 times k. And the plus, distributing the 2 to the next term. 2 times 1 is 2. Is that equal to...
This is going to be basically a FOIL. So k distributed into the k is k squared. And then we do the plus sign. k distributed into the 2 is 2k. And then for the next term, 1 times k is equal to k.
1 times the positive 2 is going to be positive 2. So now if we look, on the left-hand side, we have a k squared. We have a k squared on the right-hand side. On the left-hand side, we have a k. On the right-hand side, we have a k.
On the left-hand side, we have a 2k. On the right-hand side, we also have a 2k. And we also have the number 2 on both the left and the right-hand side.
So we've basically showed that this is true for n equals k plus 1. And we did that based off of the assumption that it is true for n equals k. And we've checked the basis step to show that it works for at least the most simple case. And basically, that is the entire proof right there. We have now proved that this statement is true. by mathematical induction.
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