00:01
We're given a sequence.
00:03
We're asked to determine total of the sequence converges, we're asked to find the limit.
00:13
Sequence as the first term, a1, which is 2.
00:19
And we have that our occurrence relation to the sequence is a .n plus 1 equals 72 over 1 plus am.
00:30
And since we assume that the limit of this sequence exists, we have that the limit, which we'll call l, as n approaches infinity, on the limit.
00:45
Sequence, and this is the same as an plus 1, this is equal to a limit as n of purchase infinity on 72 over 1 plus am.
01:01
I'm using l2 over 1 plus limit as an if purchase infinity on a .n, this is l equal to 72 over 1 plus a limit as an if purchase infinity on a .n, this is equal to 72 over 1 plus l.
01:21
So if l equals 72 over 1 plus l, we multiply both sides by 1 plus l, we get l plus l squared is equal to 72.
01:35
And so we have that l squared plus l minus 72 equals zero.
01:51
You can factor this as l minus 8 and l plus 9 equals 0.
02:10
Have that l is equal to 8 or l is equal to negative 9 notice that we have a 1 is 2 is greater than 0 and we have that a n plus 1 is equal to 72 of the 1 plus a n and so i'm going to prove using induction that sequence a n is a positive sequence that is it's downed below by 0.
03:06
So to do this, you have the first a1 is clearly downed below by 0...