00:01
Okay, so first we need to come up with a formula for this.
00:04
So we already know that the first term is one -half.
00:08
So then we can take that previous term, one -half, and add the next one.
00:13
Two times three is six.
00:16
So basically we have one -half plus one -six.
00:20
That's a plus in there.
00:22
So we can change these denominers to be the same.
00:25
So that would be three, six is the same as one -half.
00:28
So that equals 4, 6, which is 2 thirds.
00:34
So then we take the previous two terms, which summed to be 2 thirds.
00:38
And we have to add the next term to it, which is 3 times 4.
00:42
I bet you're figuring out the pattern pretty quickly here.
00:48
Well, if not, we can continue on and see that 3 times 4 is 12.
00:54
So we want the denominator to be 12 here.
00:57
So i just multiply by 4.
01:00
So multiply the top by four.
01:05
So 8 plus 1 is 9 twelfths, which equals 3 .4s.
01:09
So i bet you're noticing the pattern.
01:11
Remember, this was n is 1, n is 2, this was n is 3.
01:17
So if you notice, if you haven't already, that the numerator is the same, and this one is one more than that.
01:25
Numerator is the same, and the denominator is one more than that.
01:28
So if we were to come up with a formula for this, 1 over 1 times 2 plus 1 over 2 times 3, plus 1 over 3 times 4, dot, dot, 1 over n times n plus 1 is going to equal.
01:51
And in the numerator, and the denominator is n plus 1.
01:56
So now we need to prove that that statement's true.
02:02
So our base case is for the first term.
02:08
N is one...