What does the same mean?
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Equality is a subtle topic.
The idea of four could just be conveyed by four dots, it could also be conveyed by.
Upper case roman numerals, lower case roman numerals or a ton of other symbols
that of been used in various times
in and places throughout the world, throughout history.
All of these mean the same thing.
They all mean four. They're all in some sense equal.
But that's just equality of, of symbols of, of shapes.
It's something much more subtle happening for sequences.
When are two sequences the same? Two sequences a and b are equal, they're
the same, if they start at the same index, which I'll call big N and coresponding
terms are the same. So that a sub n equals b sub n.
Whenever n is bigger than or equal to that starting index.
Let's see how this works out in practice.
Well, here's one sequence, a sub n. Here's the sequence that starts
with a 0th term, and is defined by the rule that its nth term is 2 to the N.
And here's another sequence,
B sub N. The sequence B sub N whose 0th term is
defined to be 1 and subsequent terms will be calculated by referring
back to previous terms so that the Nth term is twice the preceding term.
These two sequences a sub n, and b sub n, they're the same.
They're equal, but they're written down really differently.
Alright, this sequence b sub n, is defined recursively and the sequence a
sub n is just defined by a single formula in terms of n.
They both start with a term labelled 0.
And corresponding terms have the same value.
A sub zero is two to the 0 using this formula which is one.
And that's the same as b sub 0.
B sub 1, is using this for a cursive formula, twice b sub
0, b sub 0 is 1, so b sub 1 is 2 times 1 which is 2.
And that's the same as a sub 1 which is, using this formula, 2 to the first power.
And that patterns continues.
These two sequences both start with a 0 term.
And each term of a sub n is twice the preceding term.
Which is exactly the recursive definition that I'm giving for b.
So a sub a thousand equals b sub of a thousand.
A sub a million b sub of a million.
A sub anything equals b sub the corresponding thing.
So the sequence a sub n and the sequence b sub n, these two sequences.
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Are equal as sequences.
Quality isn't about outside appearances, it's what's inside that matters.
It's the same for sequences. Two sequences are equal not if they've got
the same outside form, but if their corresponding terms have the same value.
You.
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