Let , ; > , be the roots of the equation x^2- 2 x- 3 =0 . Let P _n= ^n- ^n, n N . Then (11 3 -10 2 ) P ₁₀+(11 2 +10) P ₁₁-11 P ₁₂ is equal to

Step-by-step solution

aligned & x ^2- 2 x - 3 =0 _ ^ . & ^ n +2 - 2 ^ n +1 - 3 ^ n =0 & and ^ n +2 - 2 ^ n +1 - 3 ^ n =0 aligned Subtracting aligned & ( ^ n +2 - ^ n +2 )- 2 ( ^ n +1 - ^ n +1 )- 3 ( ^ n - ^ n )=0 & P _ n +2 - 2 P _ n +1 - 3 P _ n =0 aligned aligned & Put n =10 & P ₁₂- 2 P ₁₁- 3 P ₁₀=0 & n =9 & P ₁₁- 2 P ₁₀- 3 P ₉=0 & 11 ( 3 P ₁₀+ 2 P ₁₁- P ₁₁ )-10 ( 2 P ₁₀- P ₁₁ ) & =0-10 (- 3 P ₉ )=10 3 P ₉ aligned