Summation

[indigo forge]

I don’t understand how to solve this

This is from a tmua prep site not from any textbooks or past papers

[deep trellis]

discriminant

[lime cove]

er how would that work

i'd try complete the square on f(n) probs

[deep trellis]

...?

am i just be dumb or smth

[lime cove]

it tells you f(n) is a square for all values of n so it can be written as m^2 for some integer m right?
so f(n) = (n-k)^2 - k^2 + k + 12

you know the -k^2 + k + 12 must be 0 otherwise we don't have a square

[hexed merlin]

is that because
k^2-k-12 requires to have infinitely many factor pairs

f(n) = (n-k)^2 -k^2+k+12 = m^2, m∈Z
(n-k)^2 - m^2 = k^2-k-12 = C (C=k^2-k-12)
(n-k+m)(n-k-m) = C, where C is a fixed constant which has a finite number of factor pairs.
since it works for all n, there are infinitely many factor pairs which means C=0
This is how i woulda thought ab it

[deep trellis]

larping

ik yo dumbass cant think of that

[hexed merlin]

mb king ur the smartest here
if you couldn't think of that then obviously someone like me couldn't have

[indigo forge]

I don’t understand this bit sorry

Why does it have to equal 0?

[lime cove]

because (n-k)^2 is already a perfect square - adding -k^2 + k + 12 stops this from happening
if you have (n-k)^2 - m^2 is also a perfect square (where m^2 = k^2 - k - 12)
you can do some number theory to show the only solution is when m=0 but it's kinda overkill for tmua

working in mod 4 probably ?

[indigo forge]

Ohh okay I see I understood the first half but not the second half 😭😭 but yeah I’ll keep this in mind if this was to ever come up thank you marhs goat

[indigo forge]

No clue tbh, I think algebra and functions

I’m working through tyler tutoring and it’s just a worksheet on the first section

So possibly module 1?

Not too sure sorry