Mathematical Induction

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[wicked temple]

First check if it works for n=1, then show that it works for n=k+1 given the induction hypothesis that it works for n=k

[spiral summit]

ok so then $\frac{n^2+n}{2}$

[azure stream]

xNiden

[spiral summit]

$\frac{1^2+1}{2}$

[azure stream]

xNiden

[spiral summit]

$\frac{2}{2}$ = 1

[azure stream]

xNiden

[wicked temple]

For $n=1$ we have $1²+1=2$ which is divisible by $2$, hence it follows for $n=1$

[azure stream]

fäf kaka

[wicked temple]

Induction hypothesis: Let $n²+n$ be divisible by $2$ for $n=k$\
Using this we have to probe that $n²+n$ is divisible by $2$ for $n=k+1$

[azure stream]

fäf kaka

[spiral summit]

so then

$(k+1)^2+(k+1)=2$

[azure stream]

xNiden

fäf kaka

[spiral summit]

where did you get m from?

[wicked temple]

If it's divisible by 2 then there is some quotient

We don't know what m is, we just use it as unknown quotient

[spiral summit]

Ah ok i see

[azure stream]

fäf kaka

fäf kaka

[spiral summit]

hm

so then it would be $k^2+2k+1+k+1=2m$?

[azure stream]

xNiden

[spiral summit]

so then $m+1+k+1$?

[azure stream]

xNiden

[wicked temple]

How k?

[spiral summit]

idk that's what you had

[wicked temple]

See properly

[spiral summit]

you put $(k+1)^2+(k+1)=k^2+2k+1+k+1$

so then hm

[wicked temple]

You are supposed to replace k²+k by 2m

[azure stream]

xNiden

[spiral summit]

oh

[spiral summit]

what about the 2k?

[wicked temple]

It remains

[spiral summit]

so then it would just be

$m+1+k+1$?

[azure stream]

xNiden

[spiral summit]

but then we add it

making it

$m+k+2$?

[azure stream]

xNiden

[wicked temple]

[spiral summit]

im referring to this

[wicked temple]

[wicked temple]

I know

[spiral summit]

$k^2+2k+1+k+1$ = $k^2+3k+2$

[azure stream]

xNiden

[spiral summit]

m(2k+2)

[wicked temple]

Then replace k²+k by 2m

[spiral summit]

oh

[wicked temple]

How is this a product?

[spiral summit]

hold on lemme relook

oh i see

then 2m(k+2)

[wicked temple]

No

[spiral summit]

wait lemme recheck

if $m = k^2+k$ then we have $k^2+3k+2$ so then this would be 2m(2k+2)

[azure stream]

xNiden

[wicked temple]

[spiral summit]

how is it not?

[azure stream]

fäf kaka

[spiral summit]

oh i see now

so then it would be 2m+(2k+2)?

[wicked temple]

Finally

Now factor out 2

[spiral summit]

I was confused mb

alright

$2m+2(k+1)$

[azure stream]

xNiden

[spiral summit]

since its divisible by 2 then would we divide it by 2?

or would it be left like this?

[wicked temple]

2(m+k+1)

[spiral summit]

mm i see

[wicked temple]

Left like this and then with a statement that it's divisible by 2

[spiral summit]

alrighty thank you

+close