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.

Carlos or Luis? Lets get our Mexicans in order.

-con respecto

🙂

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?

@mvketelhodt You are correct. I should've spotted that.