Question Number 104544 by john santu last updated on 22/Jul/20

Given a_(n+1)  = 5a_n −6a_(n−1)   .If a_1 = 10 and a_2 = 26, find a_n ?

Answered by bobhans last updated on 22/Jul/20

let a_n = Aρ^n ⇒Aρ^(n+1) −5Aρ^n +6Aρ^(n−1)  = 0  Aρ^(n−1) {ρ^2 −5ρ+6 } = 0  ⇒ρ= 3; 2 ⇒a_n = A.3^n +B.2^n   n=1 →3A +2B = 10  n =2 →9A +4B = 26  { ((A=2)),((B=2)) :}  a_n = 2.3^n  + 2.2^n  = 2.{3^n  + 2^n  } ■

Answered by OlafThorendsen last updated on 22/Jul/20

r^2 −5r+6 = 0  (r−3)(r−2) = 0  r_1  = 2 and r_2  = 3  a_n  = λ2^n +μ3^n   To find λ and μ   we use the initial conditions.  a_1  = 2λ+3μ = 10  a_2  = 4λ+9μ = 26  ⇒ λ = 2 and μ = 2  a_n  = 2.2^n +2.3^n  = 2^(n+1) +2.3^n