POLYNOMIALS DOUBT

Not sure about the formula but if you would like another solution then here is how I solved it.

I found the roots using the quadratic formula and then using binomial theorem you can expand them. Yes, it may seem tedious but the roots are in the form of a+b and a-b so while expanding using the binomial expression a lot of terms will get cut.

I find this much easier then remembering some formula. You could give it a try maybe.

Yeah this isn’t some standard formula they teach directly, it’s something you derive, not memorize.

Start from what they gave: each root satisfies

x² = 3x − 1

Now multiply both sides by xⁿ⁻²:

xⁿ = 3xⁿ⁻¹ − xⁿ⁻²

This is true for both α and β. So:

αⁿ = 3αⁿ⁻¹ − αⁿ⁻²

βⁿ = 3βⁿ⁻¹ − βⁿ⁻²

Add them:

αⁿ + βⁿ = 3(αⁿ⁻¹ + βⁿ⁻¹) − (αⁿ⁻² + βⁿ⁻²)

So:

Sₙ = 3Sₙ₋₁ − Sₙ₋₂

That’s it. No need to learn it, just remember the trick:

if roots satisfy a quadratic, you can always build a recurrence for powers like this.