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Why am I dong any of this?

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Sequences are useful for a variety of
reasons.

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For starters, sequences help us understand
repetitive processes, and some of

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those repetitive processes are useful if
we're trying to compute something.

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Well here's an example of such a process.

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I'll define a sequence recursively, X sub
n plus 1, we want

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over X of n, plus X of n over 2.

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Maybe I'll start with the first term of
this sequence, as just being 1.

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Lets start with X of 1 equals 1, and see
what we get.

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Well X of 2 is 1 over X of 1, which is 1,
plus 1 over 2,

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that's 3 halves.
We could also try to calculate X of 3.

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I get that by taking 1 over X of 2.

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So 1 over 3 halves, and adding that to 3
halves over 2.

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Now to do that calculation, or I can write
1 over this fraction as 2 3rds.

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And instead of writing 3 over 2 divided by
2, I'll write that as 3 4ths.

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.

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I'll put this over a common denominator of
12.

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So 2 3rds

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is 8 12ths, and 3 4ths is 9 12ths.

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So all together X sub 3 is 17 12ths.

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We can compute more terms with the help of
a computer.

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Here's the X sub 2 term that we just
calculated.

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You can compute the next term, the X and 3
term is 17 12ths.

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The X and 4 terms is 577, 408ths.

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Here's the X and

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5 term, in which you'll notice is that
these terms

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are getting closer and closer to the
square root of 2.

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Even X of 3, which is 17 12ths, is close
to the square root of 2.

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Lets see how that works.

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So I want to try to convince you that

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17 12ths, is approximately the square root
of 2.

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Well if I square both sides what do I get?

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I'm getting that 17 squared divided by 12
squared.

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Should be approximately 2.
And what's 17 squared?

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Well 17 squared is 289 and 12 squared is
144.

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And is 289 over 144 close to 2?

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Yeah, because the numerator is just about
twice the denominator.

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So one way to think about the square root
of 2, is by thinking about the sequence.

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If you want to find a really good
approximation of the square root of

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2, all you've gotta do is go far enough
out in this sequence.

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