00:01
In this problem, we need to assume that the given sequence converges and we need to find its limit.
00:06
Now, let us assume that the limit as n tends to infinity of a -n is equal to the number l.
00:12
That is, let us assume that the sequence converges to the number l.
00:17
So the limit as n tends to infinity of a -n -plus -1, that will also be l.
00:24
So we are given the definition of the sequence recursively, and we have that a n plus 1 is equal to 12 minus root over a .n.
00:35
So we can take the limit as n tends to infinity on both sides of the equation.
00:41
So you get the limit as n tends to infinity of a .n plus 1 is equal to the limit as n tends to infinity of 12 minus root over a .n.
00:49
So we can write that as the limit as n tends to infinity of 12 minus the limit as n tends to infinity of root over a .n.
00:57
We can use the difference property of limits to write this.
01:00
Now, this first limit is going to be equal to 12.
01:05
That is because the limit of a constant is that constant.
01:09
And here we can write root over the limit as n tends to infinity of a .n.
01:18
This is because the limit of the square root is equal to the square root of the limit.
01:22
So this is the limit as n tends to infinity of a .n plus 1.
01:28
And now we can use these two.
01:32
We know that these two limits are l, and using them, we will obtain that l is equal to 12 minus root over l.
01:40
So we need to find the value of l that is our required limit.
01:44
So let us rearrange this equation to get root over l is equal to 12 minus l.
01:51
And now let's take the square of both sides.
01:53
So we have l is equal to 12 minus l whole square.
01:57
So that's 12 square minus 2 times 12 times l, so that's 24l plus l square.
02:05
So what we obtain is l square minus 24l minus l, so that's minus 25l plus 144, that is equal to 0.
02:14
We can rearrange the equation like this.
02:17
Then we can write l square minus 9l minus 16l plus 144 is equal to 0.
02:25
This is because negative 9 and negative 16, these two numbers add up to negative 25 and their product is 144.
02:34
So this is what we have...