00:01
We are given a1 equals 1.
00:03
First, let's show a .n is always bigger than equal to 1 for all n.
00:15
So to do this, let's use induction.
00:21
We have our base case, which is n equals 1.
00:28
So we have a1 equals 1.
00:30
That's given up here.
00:32
So that's bigger than or equal to 1.
00:34
So that's a true statement.
00:37
Now we go to the inductive step.
00:43
This is where we suppose a .n is bigger than or equal to 1 for some n.
00:54
So little n can be any number.
00:57
Now we'd like to show, want to show, if we increase n by 1, this is the inductive step, that it's still bigger than are equal to 1.
01:12
So to show this, let's just use the formula for a .n.
01:17
Plus 1 that's given above.
01:19
Circle in red.
01:23
3 minus 1 over a .n.
01:28
However, what do we know about an, that this is bigger than or equal to 1? so in this case, this is bigger than or equal to 3 minus 1 over 1, because the most we could subtract is when the denominator is smallest, the denominator is smallest when an is 1.
01:49
This equals 2, and that is bigger than or equal to 1.
01:54
So by induction, we have an bigger than or equal to 1, so let's keep that in mind and go on to the next page.
02:06
So i'll just record that up here.
02:08
We just showed an bigger than or equal to 1.
02:13
Now let's show an, the sequence is monotone increasing.
02:27
So here we let's do also use induction, very useful technique.
02:35
We have our base case again.
02:38
So here we like to show a1 is less than a2.
02:42
So a1 from the previous page was equal to 1.
02:46
A2 if you use the formula, so 3 minus 1 equals 2.
02:52
So we see that yes, this is a true statement, 2 is bigger than 1, so the base case holds.
03:01
Now we go to the inductive step.
03:09
So here we will suppose, here's our inductive hypothesis for some n.
03:24
So n can be any number but it's just one particular fixed value.
03:28
Now we want to use this to show, want to show that if you increase n by one, so here this n becomes n plus one.
03:39
And then on the right hand side, add one more to this end and add those ones together.
03:46
This is what we'd like to show.
03:48
So let's go ahead and do this.
03:51
A .n plus 2, 3 minus 1 over a .n plus 1.
03:57
Now we can go ahead and write the following, and i'll explain why.
04:02
The reason we can do this is by our hypothesis so if you just go ahead and take this equation this inequality in blue if you go ahead and do some algebra we can rewrite this and then multiply both sides by a negative so this means that if you have the negative if you have the plus one on the end it's a larger fraction so here we replaced a larger fraction with a smaller fraction so so due to this inequality right here, this justifies this.
04:43
And now by using the formula on the previous page, we know this is equal to an plus 1.
04:50
So we started with n plus 2.
04:52
We have bigger than are equal to and then am plus 1.
04:57
So this proves by induction.
05:03
So by induction we conclude that this sequence an is monotone increasing.
05:18
So that's our second fact...