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Prove that the square of an odd number is of the form 8q + 1.
Let m = an odd number
All numbers can be represented in the form 4k ± 1.
Hence, m = 4k ± 1.
m² = (4k ± 1)² = 16k² ± 8k + 1 = 8k(2k ± 1) + 1
But what if you choose 2k ± 1? Won't that give the same answer?
m² = (2k ± 1)² = 4k² ± 4k + 1 = 4k(k ± 1) + 1
Huh? Now it's only of the form 4q + 1???
Uh...just one more question, how can ±x + ±x = 0?
Thanks luca!
Because by looking at the first equation, you know that you will square root both the top p and the bottom q. Can't this be directly written also as
? Because you have the choice to sqrt them separately?Also, +-'s are a major headache when it comes to the distributive law! How is it supposed to work?
...What happens if you multiply
by ... or by ?Also... is
????I'm a bit confused about the +- sign.
If
Then
Do the +-'s cancel out or what???
If you have this, then multiply
to both sides, where does the +- go? Does it stay with the or go to the other side?? yea im a bit confusedI think what's "cool" changes as you grow up. In Junior School, it's cool to be fascinated in sciences, in Mid school it's not cool to be smart, in High school smart people get to make fun of others.
Oh I see, thanks then
Let's just start at 67116436219957, shall we?
Hmm, i will have to mull over that...
In the meantime, I came up with this:
where
is the nth term in the first table I provided.The proving is incomplete... but it kinda implies that
, soIt might work on the other table too... dunno, and it breaks down when you get into tricky relations
Can you please check / improve this thx:)
311311222113111231131112132112311321322112111312211312111322212311322113212221
132113213221133112132113311211131221121321131211132221123113112221131112311332111213211322211312113211
How is the first one obvious though? Is there a non-graphical way to do this? I think there probably is... we did something kinda similar in physics, but I can't get my mind around it.
Trebuchet!
Awgennel
a 1 2 3 4 5
b 3 0.75 0.33 1.875 0.12
what is the quickest and fastest way of finding the rule between relations like this???
and this
m 1 2 3 4 5
n 7.2 5.09 4.16 3.6 3.22
They are pretty simple in that they will be only be in the form
, but the book gives this profusely long trial error 'draw graphs and see if it is a straight line' approach. ThxNooooo too many in a single post!!!
223241
47 - 四十七 (Chinese)
siebzehn
So if there is no
infront of the square root, it means it is positive?Oh I see, too bad the 'solve' command only gives 2 solutions >.<
Global Warming? Will it happen? What are we going to do about it?
This guy breaks down the argument very simply by analysing a grid of scenarios, and his own argument is quite strong.
tsk tsk microsoft trying to rule the world again... as shocamefromebay said, it looks too much like a calculator.
Nope identity, you gotta add one more.
But how???
Perhaps the ball merely retained some of its initial velocity. Using that kind of speed meter in the car can lead to major inaccuracies.
Is the answer to 63 . All the other questions on the test are hard! You must have to be a polymath to do well on that.
But it's impossible that it just drops to the ground, because first it must accelerate to 100kmh-¹, right?