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# Inductive reasoning

## Video transcript

Jill looked at the following
sequence. 0, 3, 8, 15, 24, 35. And it just keeps going, I
guess, with a dot, dot, dot. She saw that the numbers
were each 1 less than a square number. 0 is 1 less than 1, which
is a square number. 3 is 1 less than 4. 8 is 1 less than 9. 15 is 1 less than 16. Yeah, they were all 1 less
than a square number. And conjectured that the
nth number would be n squared minus 1. Now conjecture, that sounds
like a very fancy word. When someone makes a conjecture,
they conjecture, that just means that they're
making a statement that seems, or they're making a proposition
that seems likely to be true. It seems like a very reasonable
thing to say. But it's not definitely true. So she conjectured that the
nth number would be n squared minus 1. The reason why this is a
conjecture as opposed to a 100% definitely true statement,
is we don't know whether this pattern
continues. She's just going off of the
pattern that she saw so far and she just generalized it. She just assumes that
it keeps on going. But we don't know whether it
necessarily keeps going. Maybe the next number, you would
expect it to be 48, but maybe it's not 48. Maybe it's something weird. Maybe it's 2. Maybe it's 500. And so the conjecture wouldn't
hold up if you were to see that, but based on the evidence
you see so far it seems completely reasonable
that this pattern would continue. And so she conjectured that
the nth number would be n squared minus 1. Completely reasonable. Now did Jill use inductive
reasoning? Yes, she used inductive
reasoning. That's what inductive
reasoning is. You see a pattern. In this case, every term in this
sequence so far was-- if it's the third term, it
was 3 squared minus 1. The fourth term is 4
squared minus 1. The fifth term is 5
squared minus 1. So she saw the pattern and she
just generalized it to say, well, I think or I've
conjectured that the nth number will be n squared
minus 1. That's what inductive reasoning
is all about. You're not always going to be
100%, or you definitely won't be 100% sure that you're right,
that the nth number will be n squared minus 1. But based on the pattern you've
seen so far, it's a completely reasonable thing
to-- I guess you could say-- to induce.