sigma

[wary meadow]

Help! I have no clue how to solve this problem, our teacher doesn't know how to teach. I just recently found out about properties of sigma notation on youtube and I HAVE ABSOLUTELY NO CLUE HOW TO SOLVE THIS

[twilit needle]

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[wary meadow]

The answer is 9, our teacher wants us to solve it and prove that the answer is 9

[nova rapids]

okay

we can start by rationalizing the denominator first

[wary meadow]

Shit

[nova rapids]

?

[wary meadow]

Fk fk I skipped highschool algebra

[nova rapids]

so do you know or do you not know?

[wary meadow]

Wait holdon the organic chemistry teacher has a tutorial for it

[nova rapids]

i can teach you here

it isn’t that hard

[wary meadow]

For real? I'm not being a bother to you or sumthin

Lemme get a pen rq

[nova rapids]

no lol

why am i doing this in the first place 💀

anyways

[wary meadow]

Okay start

[nova rapids]

let’s propose we have an irrational number $\sqrt{a}+\sqrt{b}$

[supple mauve]

;( | 追放された興奮

[wary meadow]

I have a pen and a paper

[nova rapids]

as in the equation

[wary meadow]

Mhm

[nova rapids]

to rationalize the denominator

[orchid sparrow]

Telescoping series!

[nova rapids]

we have to multiply by its conjugate, or $\sqrt{a}-\sqrt{b}$

[supple mauve]

;( | 追放された興奮

[nova rapids]

no shit 💀

[wary meadow]

Its opposite?

[orchid sparrow]

But yes, rationalize like how this dude is saying

[nova rapids]

discord crashed

sorry

[wary meadow]

So sqrt(i + 1) + sqrt(i) multiplied by sqrt(i+1) - sqrt(i)?

[nova rapids]

yes

butt

multiply on the numerator and denominator

so we can have a factor of one

[wary meadow]

By that you mean?

[nova rapids]

[wary meadow]

Sorry im a bit of a dumbass

[nova rapids]

it’s okay

we multiply the given fraction

[wary meadow]

Does this look right

[nova rapids]

by $\frac{\sqrt{i+1}-\sqrt{i}}{\sqrt{i+1}-\sqrt{i}}$

[supple mauve]

;( | 追放された興奮

[nova rapids]

right concept, but look ^

[wary meadow]

OHHHH

[nova rapids]

multiply by this

because we get factor of 1

[wary meadow]

Right

And then

[nova rapids]

gotchu

an irrational multiplied by its conjugate is equal to a-b

so in this case

denom would be i+1-i

which is?

[wary meadow]

Do I just straight up multiply them?

There is two sqrt(i + 1)

Would that make it (sqrt(i+1))²

[nova rapids]

yep

same with sqrt(i)

in general though

[wary meadow]

Wait wait

The radical cancel out right

[nova rapids]

$(\sqrt{a}+\sqrt{b}) (\sqrt{a}-\sqrt{b})=a-b$

[supple mauve]

;( | 追放された興奮

[nova rapids]

yep

math is magic

[wary meadow]

Im getting close

I can feel victory

[nova rapids]

haha

yes

[nova rapids]

now we have $\sum_{i=1}^{99} (\sqrt{i+1}-\sqrt{i})$

[supple mauve]

;( | 追放された興奮

[nova rapids]

this is telescoping

[wary meadow]

Now what how do I use the properties of sigma notation on this

[wary meadow]

Yes but there are radicals

[nova rapids]

you don’t

see how it is i+1 compared to i

let me write out a few terms of the sum to show you

[wary meadow]

Okay

[nova rapids]

sqrt(2)-sqrt(1)+sqrt(3)-sqrt(2)+sqrt(4)-sqrt(3)...sqrt(100)-sqrt(99)

see how the sqrt(2) and sqrt(3) cancel in this sum?

because there is a negative and positive part for it?

[wary meadow]

I have completely no idea why they cancel out

[nova rapids]

look at sum again

and look for coefficients of sqrt(2) and sqrt(3) terms

[wary meadow]

Do you mean the sqrt(2) and negative sqrt(2)

Or am I being dumb

[nova rapids]

yes, sure

sqrt(3) also has a negative component in the sum

[wary meadow]

Yeah

[nova rapids]

now u see how they cancel right

[wary meadow]

😔

Sorry but my mind cant process how sqrt(2) and sqrt(3) cancel out

ODJWPZJEKAKJF

Im hopeless

Lets just uhh skip that part

Im too much of a numskull

[orchid sparrow]

You can't proceed if you don't notice how the terms cancel out....

[nova rapids]

bro

[orchid sparrow]

Look, there is sqrt(2) and -sqrt(2)

Add them.

[nova rapids]

that’s the whole concept of a telescoping series 💀

[wary meadow]

Yeah thats what I meant

[orchid sparrow]

Yh

[wary meadow]

Here

[nova rapids]

so what is sqrt(2)-sqrt(2)

[orchid sparrow]

Yeah

[wary meadow]

0

[orchid sparrow]

Yes, those 2 terms

[nova rapids]

what about sqrt(3)-swrt(3)

[orchid sparrow]

Same goes for sqrt(3) and ... so on

[wary meadow]

Fk I was just misunderstanding you

[nova rapids]

looooool

[wary meadow]

Yup

[nova rapids]

so this happens

for all of the numbers in the series

except for the first and last term

[wary meadow]

Mhm

[nova rapids]

i.e. -sqrt(1) and sqrt(100) in this case

[wary meadow]

OOOOOOO

[orchid sparrow]

Just observe those pairs...they all cancel out except for the first and last term, they dont have their respective negative factors.

[wary meadow]

I can see that

[nova rapids]

so $\sum_{i=1}^{99} (\sqrt{i+1}-\sqrt{i})=\sqrt{100}-\sqrt{1}$

[supple mauve]

;( | 追放された興奮

[wary meadow]

THANK YOU SO MUCH

[orchid sparrow]

So yeah, this is what you get.

[wary meadow]

That explains why the answer is 9

[orchid sparrow]

And that's just 10-1=9

[orchid sparrow]

Yup

[wary meadow]

Thank you for sticking with my numskull ass

[nova rapids]

i have had worse people

[wary meadow]

https://cdn.discordapp.com/emojis/859795873915207721.gif?quality=lossless&size=48

[orchid sparrow]

Yeah same lol

You did a great job grapsing the concepts.

[wary meadow]

How do I develop my problem solving skills like y'all so I can be less of a burden

[orchid sparrow]

Practice?

And focus on improving your observation skills

[wary meadow]

Okay I'll try to solve as many of these as I can on my free time

Thank you again!

[nova rapids]

spam practice

[wary meadow]

+close