00:01
We now determine an explicit formula for the sequence defined recursally that satisfied the conditions a sub 0 equals 2 and a sub n equals a sub n minus 1 divided by 2 for n equals 1, 2 3 and so on.
00:17
So first let's use this condition a sub n equals a sub n minus 1 divided by 2.
00:23
I'm going to plug in the values of n equals 1, 2, 3 and so on to determine the next few terms.
00:30
We plug in n equals 1 into this condition i get a sub 1 and this equals a sub 1 minus 1 divided by 2 and this equals 1 minus 1 is 0 so this becomes a sub 0 divided by 2 we are already given the value of a sub 0 that is 2 so if we replace a sub 0 this will become 2 by 2 and this equals 1 the same way let's plug in n equal to 2 into that uh conditions so we are going to get a sub 2 and this equals a sub 2 minus 1 divided by 2 so this equals 2 minus 1 is 1 so this becomes a sub 1 divided by 2 and this equals a sub 1 we determine it is 1 so this will be 1 divided by 2 now let's also plug in n equal to 3 and so we get a sub 3 and this equals a sub 3 minus 1 divided by 2.
01:33
So this becomes a sub 2 divided by 2.
01:37
So this equals a sub 2 is 1 by 2.
01:41
This is divided by 2 and this will become 1 by 2 times of 1 by 2.
01:47
That equals 1 by 4.
01:53
Okay, now that we have determined the terms, i'm going to write the sequence defined recursively, starting from a sub 0.
02:02
So first a sub 0, then a sub 1, a sub 2, and a sub 3.
02:11
We are told a sub 0 equals 2 and we determine a sub 1 equals 1, a sub 2 equals 1ā2 equals 1ā2 and then a sub 3 equals 1 4th.
02:28
Now let's try to understand this.
02:31
You can see that every term is molecular...