Sequences problem I can’t solve

[azure ivy]

What are the variations of Pn(X) in [0, 1]?

[halcyon prism]

X^n + X^(n+1)+…+X^2+X-1
Did you mean X^(n-1)?

@zinc vessel

[zinc vessel]

yes

mb

[short glen]

lemme think about it...

give me some time

yeah... i think that i got it

lemme show you

i don't know how to use TexIt, but i will try my best

$let p(x) = x + x^2 + x^3 + ... + x^n, (i will talk about the -1 later)$

[winged hare]

Waleed Al-Thqfi

[short glen]

so let's do some algebra
$p(x) = x(1 + x + x^2 + ... + x^(n-1))$

[winged hare]

Waleed Al-Thqfi

[halcyon prism]

$P_n(x)=-2+\sum_{k=0}^n x^k$

[winged hare]

SWR

[short glen]

$p(x) = x(1 + p(x) - x^n)$
$p(x) = x + x*p(x) - x^(n+1)$

[winged hare]

Waleed Al-Thqfi

[short glen]

$p(x) - xp(x) = x - x^{n+1}$

[winged hare]

Waleed Al-Thqfi

[short glen]

$p(x)(1-x) = x-x^{n+1}$,
$p(x) = {(x-x^{n+1})}/{(1-x)}$

[winged hare]

Waleed Al-Thqfi

[short glen]

now.. let $p(x) = 0$

[winged hare]

Waleed Al-Thqfi

[short glen]

$0 = x-x^{n+1}$, x not equal to 1

[winged hare]

Waleed Al-Thqfi

[short glen]

let's take n = 6 as an example
$0 = x-x^{7}$, n + 1 = 7

[winged hare]

Waleed Al-Thqfi

[short glen]

$0=x(1-x^{6})$

[winged hare]

Waleed Al-Thqfi

[short glen]

$0=x(1+x)(x^2-x+1)(1-x)(x^2+x+1)$

[winged hare]

Waleed Al-Thqfi

[zinc vessel]

this feels overcomplicated, it says to deduce not prove

[short glen]

i fcked it up, right ?

[zinc vessel]

think so

[short glen]

deduce? i don't know what does that means

[zinc vessel]

pretty sure you're supposed to use the inequality to show that the sequence Un+1 < Un therefore its decreasing

[short glen]

yeah we find 7 values of x for n = 6, but one of them is 1, so we assumed from the beggining that x doesn't equal to 1

[short glen]

i think i read your question wrongly, my bad

[zinc vessel]

no worries

[short glen]

but i think i take it much crazier to find all the solutions for x+x^2+...+x^n in one signle formula lmao

[zinc vessel]

lmao

we do have a solution tho, Pn(Un) = 0

[halcyon prism]

@zinc vessel do you still need help?

[zinc vessel]

yes

[halcyon prism]

@zinc vessel can you use calculus?

[zinc vessel]

yes