00:01
In this problem, we're being asked to find the sum of the first n even integers in part a, and in part b we have to prove that this formula indeed works.
00:14
So part a, first we have to find a formula for the sum of the first n even integers.
00:24
So we know that we're going to have a summation here, and we can say i equals 1 up to n, and an even number is defined as the number of the form 2, in this case we'll say i, where i is an integer.
00:44
So a few examples that will help us try to figure out a formula for this, let's say when n equals 4, then we have 2 times 1 plus 2 times 2 plus 2 times 3 plus 2 times 4.
01:07
And if you notice here, we can factor out the 2, so what remains is, what we're left with inside is 1, here i'll highlight it under red, so again i'm just factoring out the common multiple of 2 here, and then what we're left with is not on the line, so we have 1 plus 2 plus 3 plus 4, and we can write 1 plus 2 is 3, plus 3 is 6, plus 4 is 10, so 2 times 10 is 20.
01:49
So in this case, when n equaled 4, we got a value of 20, and we know that 20 equals 4 times 5, okay that might be useful.
02:05
So now let's examine the example when n equals 5, so again i can write out as i did before, 2 times 1 plus 2 times 2, but it's going to be the same thing, all that's left that's different is that i'm now adding 2 times 5, so in other words, underneath here, this will still be 10, but now we're adding 5, so that's 2 times 15, which is 30.
02:42
And again, if i underline this, when our example, when our n is 5, we have 30, and we can see that 30 equals 5 times 6, and if you practice out a few more examples, you'll notice that the case is that the summation is actually equal to whatever n you pick, times n, oops, times n plus 1.
03:16
If i could write down, there you go, so in part a, we were asked to find a formula for the summation of the first n even terms, and we found now that the formula is n times n plus 1.
03:37
So now we have to go about proving this.
03:40
Well, this form of proof, we're going to be using something called induction, so in this case, our proposition for the proof is that for any n, for any n as an integer, or yeah, that works, no, no, no, as a natural number, we're saying that the sum, when i starts at 1, up to n, of 2i, it's going to equal n times n plus 1.
04:28
So now we have to go about proving this.
04:30
So in our inductive, or in our proof, we have to first start with the base case.
04:36
We have to prove that the proposition is true for our base case, and our base case for this case is when n equals 1.
04:48
So when n equals 1, then our summation, when we plug in n for 1, oops, i equals 1, 2i, well, we're just going to add 2 times 1, which is 2.
05:10
And now when we use our formula, we do 1 times 1 plus 1, which is 1 times 2, which is 2.
05:22
And as you can see, 2 equals 2.
05:25
So yes, the base case is proven.
05:28
Now we move on...