Let α, β are the roots of the quadratic equation 2 x2-5 x+1=0. If Sn=(α)2 n+(β)2 n then find the value of (4 S 2021+ S 2019/ S 2020).

Question

Let α, β are the roots of the quadratic equation 2 x2-5 x+1=0. If Sn=(α)2 n+(β)2 n then find the value of (4 S 2021+ S 2019/ S 2020). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Given α, β are the roots of the quadratic equation 2 x2-5 x+1=0 Let us find an equation with roots α2 and β2, let y=x2, so x=√y 2 y -5 √ y +1=0 ⇒ 2 y +1=5 √ y ⇒ 4 y 2+4 y +1=25 y <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/6d07b9051c55ec60210662dc2a46e468-.png" /> Put α2= c and β2= d Now, S n =( c ) n +( d ) n Consider 4 S 2021+ S 2019 =4( c 2021+ d 2021)+ c 2019+ d 2019 = c 2019(4 c 2+1)+ d 2019(4 d 2+1) = c 2019(21 c )+ d 2019(21 d ) =21 S 2020 Hence (4 S 2021+ S 2019/ S 2020)=21

Q. Let α,β are the roots of the quadratic equation 2x2−5x+1=0. If Sn​=(α)2n+(β)2n then find the value of S2020​4S2021​+S2019​​.

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