inequality induction

[fluid iron]

help i dont get how to do the inductive step part
i've got it up to assume n=k and then idk how to do the rest HELP 🙏

[mental moth]

what do you not understand?

do you not know the method or smth?\

youve got to do n=k+1 AFTER the n=k then try to solve someway to get an inequality greater than the other one

lemme show u

[fluid iron]

thanks 🙏

[rough pollen]

k! > k(k+1) (assumption)

Inductive case:
you have to show that
(k+1)! > (k+1)^2 + (k+1)

(k+1)! = (k+1) * k! > k(k+1)(k+1)
(k+1)! > k(k+1)(k+1)
To show that (k+1)! > (k+1)^2 + (k+1)
Then,
show that
k(k+1)(k+1) > (k+1)^2 + (k+1)

k^3+k^2-2k-2>0

then i subbed in k=-1
and that that
(-1)^3 + (-1)^2 -2(-1) -2 = 0
so then by the factor theorem
k+1 is a factor
by inspection
it factorises to

(k+1)(k^2-2) > 0

k>=4 so
this inequality will Always hold true
so you proved it

idk if im right but thats the algebra side of the proof

[fluid iron]

OKAY i see i will write it out but it does make sense to me in my head now

thank you

[rough pollen]

nw

[mental moth]

bruh

i was about to send my solution

[mental moth]

yh thats pretty much right

[fluid iron]

send send send

[mental moth]

or you could do by comparison of LHS and RHS by expanding factorial and then subbing in the expression for n=k then expanding out further both sides and factorising to show one side is greater than the other by logic

[mental moth]

alr

conclusion you already know

so i just couldnt be asked

i should have also put a line that LHS > RHS

[fluid iron]

love this ty

do you do ocr?

[mental moth]

yh ocr a

[fluid iron]

do you have predictions for tomorrow

[mental moth]

nws

[mental moth]

wdym?

[fluid iron]

anything you think will come up in the paper?

[mental moth]

nah its so spontaneous

anything could pop up

[fluid iron]

ocr tests everything every year

[mental moth]

yh

[fluid iron]

theres no difference between pure 1 and pure 2 right

[mental moth]

u can predict paper 2 after u do paper 1

[fluid iron]

yeah

what optionals do you do

[mental moth]

yh but if u dont get something in paper 1, you will defo get it in paper 2

[mental moth]

mech and stats

[fluid iron]

yes yes that's true

[mental moth]

stats is fine but i gotta revise a lot for mech

[fluid iron]

good luck

[mental moth]

wbu?

[fluid iron]

when's mech

[mental moth]

thanks u too

[fluid iron]

i do stats and add pure

[mental moth]

3 weeks

[mental moth]

thats so lucky ibr

[fluid iron]

nahh add pure is so hard

but icl id rather do that than mech i dont like mech

so maybe yeah

[mental moth]

yh but once u get used to mech its fine ibr