Let , are the roots of the quadratic equation 2 x^2-5 x+1=0 . If S_n=( )^ 2 n +( )^ 2 n then find the value of 4 S₂₀₂₁+S₂₀₁₉ S₂₀₂₀ .

Step-by-step solution

Given , are the roots of the quadratic equation 2 x^2-5 x+1=0 Let us find an equation with roots ^2 and ^2 , let y=x^2 , so x= y 2 y-5 y +1=0 2 y+1=5 y 4 y^2+4 y+1=25 y 4 y^2-21 y+1=0 _d^c . Put ^2=c and ^2=d Now, S_n=(c)^n+(d)^n Consider aligned 4 S₂₀₂₁+S₂₀₁₉ & =4 (c²⁰²¹+d²⁰²¹ )+c²⁰¹⁹+d²⁰¹⁹ & =c²⁰¹⁹ (4 c^2+1 )+d²⁰¹⁹ (4 d^2+1 ) & =c²⁰¹⁹(21 c)+d²⁰¹⁹(21 d) & =21 S₂₀₂₀ aligned Hence, 4 S₂₀₂₁+S₂₀₁₉ S₂₀₂₀ =21