Question

Let $\alpha, \beta$ are the roots of the quadratic equation $2 x^2-5 x+1=0$. If $S_n=(\alpha)^{2 n}+(\beta)^{2 n}$ then find the value of $\frac{4 S _{2021}+ S _{2019}}{ S _{2020}}$.

Answer 1

Given $\alpha, \beta$ are the roots of the quadratic equation $2 x^2-5 x+1=0$
Let us find an equation with roots $\alpha^2$ and $\beta^2$, let $y=x^2$, so $x=\sqrt{y}$
$2 y -5 \sqrt{ y }+1=0 \Rightarrow 2 y +1=5 \sqrt{ y } \Rightarrow 4 y ^2+4 y +1=25 y $

Put $\alpha^2= c$ and $\beta^2= d$
Now, $S _{ n }=( c )^{ n }+( d )^{ n }$
Consider
$4 S _{2021}+ S _{2019} =4\left( c ^{2021}+ d ^{2021}\right)+ c ^{2019}+ d ^{2019} $
$ = c ^{2019}\left(4 c ^2+1\right)+ d ^{2019}\left(4 d ^2+1\right)$
$ = c ^{2019}(21 c )+ d ^{2019}(21 d )$
$ =21 S _{2020}$
Hence $\frac{4 S _{2021}+ S _{2019}}{ S _{2020}}=21$