Why am I dong any of this?
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Sequences are useful for a variety of reasons.
For starters, sequences help us understand repetitive processes, and some of
those repetitive processes are useful if we're trying to compute something.
Well here's an example of such a process.
I'll define a sequence recursively, X sub n plus 1, we want
over X of n, plus X of n over 2.
Maybe I'll start with the first term of this sequence, as just being 1.
Lets start with X of 1 equals 1, and see what we get.
Well X of 2 is 1 over X of 1, which is 1, plus 1 over 2,
that's 3 halves. We could also try to calculate X of 3.
I get that by taking 1 over X of 2.
So 1 over 3 halves, and adding that to 3 halves over 2.
Now to do that calculation, or I can write 1 over this fraction as 2 3rds.
And instead of writing 3 over 2 divided by 2, I'll write that as 3 4ths.
.
I'll put this over a common denominator of 12.
So 2 3rds
is 8 12ths, and 3 4ths is 9 12ths.
So all together X sub 3 is 17 12ths.
We can compute more terms with the help of a computer.
Here's the X sub 2 term that we just calculated.
You can compute the next term, the X and 3 term is 17 12ths.
The X and 4 terms is 577, 408ths.
Here's the X and
5 term, in which you'll notice is that these terms
are getting closer and closer to the square root of 2.
Even X of 3, which is 17 12ths, is close to the square root of 2.
Lets see how that works.
So I want to try to convince you that
17 12ths, is approximately the square root of 2.
Well if I square both sides what do I get?
I'm getting that 17 squared divided by 12 squared.
Should be approximately 2. And what's 17 squared?
Well 17 squared is 289 and 12 squared is 144.
And is 289 over 144 close to 2?
Yeah, because the numerator is just about twice the denominator.
So one way to think about the square root of 2, is by thinking about the sequence.
If you want to find a really good approximation of the square root of
2, all you've gotta do is go far enough out in this sequence.
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