Inductive reasoning 3 | Sequences, series and induction | Precalculus | Khan Academy

Inductive reasoning 3 | Sequences, series and induction | Precalculus | Khan Academy


Luis looked at the expression x
minus y times x squared plus xy plus y squared, and
wrote the following. He said, using the distributive
property– so he took x minus y and he
distributed this expression onto the x and this expression
onto the negative y. So that’s why we have an
x times this entire expression over here. x times the entire expression
minus y times this entire expression. Then he says, using the
distributive property again, this is equal to– so then he
distributed this x into that and got all of the first
three terms here. He got these first three
terms. And then he distributed the y. He distributed the y over here
and he got these three terms. And then he saw, looks like
that this term x squared y cancels out with a minus x
squared y, and the xy squared cancels out with the negative
xy squared. Then he saw that it equals
x to the third minus y to the third. And he wrote down, I conjecture
that x minus y times x squared plus xy plus y
squared is equal to x to the third minus y to the third
for all x and y. Did Carlos use inductive
reasoning? Explain. Well, inductive reasoning is
looking at a sample of things– looking at some set
of data, if you will– and then making a generalization
based on that. You’re not 100% sure, but based
on what you’ve seen so far, you think that the pattern
would continue. Or you think it might be true
for all things that have that type of property or whatever. Now in this situation,
he didn’t look at some type of a sample. He actually just did a proof. He multiplied this out
algebraically. In fact, it’s incorrect for him
to say I conjecture here. A conjecture is a statement or
proposition that is unproven, but it’s probably going
to be true. It’s unproven but it seems
reasonable, or it seems likely that it’s true. This isn’t a conjecture. This is proven. He proved that x minus y times
x squared plus xy plus y squared is equal to x to the
third minus y third. He should have written– and
this is a much stronger thing to say– he should have said, I
proved that this is true for all x and y. So to answer the question, did
he use inductive reasoning? No. I would say that he made
an outright proof. No, he made a proof. Inductive reasoning would have
been, if he would have saw, if you would have given him 5 minus
2 is equal to– or 5 minus 2 times 5 squared plus 5
times 2 plus 2 squared, and you saw that that was equal to
the same thing as 5 to the third minus 2 to the third. And then let’s say he did it
for, I don’t know, one in seven in a couple of examples. And it kept holding for all the
examples that it was the first number cubed minus the
second number cubed. Then it would have been
inductive reasoning to say that that is true for
all numbers x and y. But here it’s not
an induction. He didn’t use induction. Or I shouldn’t say induction. He didn’t use inductive
reasoning. He outright proved that
this statement is true for all x and y.

4 thoughts on “Inductive reasoning 3 | Sequences, series and induction | Precalculus | Khan Academy

  • To expand a bit, the method of Mathematical induction, is the technique of proving a statement, P(n), where P(n) is some mathematical statement, by first proving that P(n) holds true for some base value (say, 0, or 1) and then, proving that P(n+1) is true, whenever P(n) is true. In other words, if we assume that P(n) is true, we then set out to prove that P(n+1) is also true, or P(n) -> P(n+1).

  • no, x should be the inverse expression of y and be put into that while the y's should be distribute proportionately throughout the formula insofar causing the x the third to be redistributed along the logical but exponentially true but hypothetical reasoning unproven but reasonable in the state of y's and proving x the third is inductive reasoning..comprende?

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