Hello, I'm having trouble understanding what this is this trying to say.

[magic lake]

This is pretty easy to some but for me I'm sort of struggling on it. Math was always my weak point so if someone could help, that would be of great appreciation

[sage ibex]

Well, just do what the assignment says. Write down the squares of two consecutive, positive integers, e.g. 3,4 -> 3²+4²=9+16=25.
When you're done doing that, try to find some kind of pattern and write that down. "Conjecture" in this context means "hypothesis".

[magic lake]

so the question is saying, write down two numbers, write down the square of those two numbers, add those two together?

[sage ibex]

And the numbers need to be natural numbers (1,2,3,4,...) that are next to each other.

[magic lake]

so you can't pick four and seven for example?

only numbers that are after one another

[sage ibex]

Exactly.

[magic lake]

right so from what i've seen, the conjecture seems to be that one of the squared numbers ends with an even number, and the other ends with an odd number

[sage ibex]

Well that is obvious since the product of even numbers will always be even while the product of odd numbers will always be odd.

[magic lake]

3²+4²=9+16=25

4² + 5² = 16 + 25

5² + 6² = 25 + 36

6² + 7² = 36 + 49

I see another pattern

but i can't really explain it

i'm not sure on how i could write down a conjucture for this

[sage ibex]

You know, writing "the sum of two squares is always odd" is already a decent conjecture, though extremely easy to prove.

[magic lake]

5² + 6² = 25 + 36
wouldn't this refute it?

wait hold on

nevermind

I appreciate your help

Makes more sense to me now

i was looking at the sum and not the final answer

do you mind if I ping you if I have any other questions?

[sage ibex]

No, not at all. There's a myriad of conjectures one could come up with here.

[magic lake]

@sage ibex do you think you could help me out with this one? I've been looking at this question for a while and i don't really understand it one bit

is it -3?

since 1 over -3 is less than a real number

[sage ibex]

There are two cases to consider: n being positive and n being negative.

If n is positive, 1/n is smaller than n if n > 1. It is the opposite if n is negative.

[sage ibex]

So this is correct.

But it is obviously not the only solution.

[outer bough]

For question A, you can solve the general inequation $\frac{1}{n} < n$ for all $n\in\mathbb{R}$ and then exclude all options that don't satisfy the solution. This can be solved fairly easily by both algebraic or geometric / graphical methods. @sage ibex has already mentioned what the solution of this inequation is.

OR...

You can test each case, as in...

For option (A), put $n = -3$: The statement $\frac{1}{n} < n \implies \frac{1}{-3} < -3$ is the same as $-0.333... < -3$ and is clearly FALSE.

Repeat for options (B), (C), (D), and (E).

[raven basalt]

Jay

[magic lake]

So subsitute each one and find which one is correct?

Yes, it all makes sense to me now

Except B

1 over 1/4 < 1/4

What's the final substition for this?

Do you flip the reciprocal?

[outer bough]

1 over 1/4 is 4, and 4 < 0.25 is FALSE.

Only (D) and (E) are true, as (C) is 1 < 1.