Question Number 104428 by bemath last updated on 21/Jul/20

given a_(n+1) =a_(n−1) +2a_n   if a_3 = 0 & a_5  = −1  find a_n

Answered by Dwaipayan Shikari last updated on 21/Jul/20

Put n=4  a_5 =a_3 +2a_4   −1=2a_4   a_4 =−(1/2)

Commented bybemath last updated on 21/Jul/20

a_n  ?

Commented bybemath last updated on 21/Jul/20

any one help me?

Answered by mathmax by abdo last updated on 22/Jul/20

a_(n+1) =a_(n−1)  +2a_n  ⇒a_(n+2)  =a_(n )  +2a_(n+1)  ⇒a_(n+2) −2a_(n+1) −a_n =0  caracteristic equation r^2 −2r−1 =0 ⇒r^2 −2r+1−2=0 ⇒  (r−1)^2 −((√2))^2  =0 ⇒(r−1−(√2))(r−1+(√2))=0 ⇒r =1+(√2)or r =1−(√2)  a_n =α(1+(√2))^n  +β(1−(√2))^n   a_3 =α(1+(√2))^3 +β(1−(√2))^3  =0 and a_5 =α(1+(√2))^5  +β(1−(√2))^5 =−1 ⇒  we get the system    { (((1+(√2))^3  α +(1−(√2))^3 β =0)),(((1+(√2))^5  α +(1−(√2))^5  β=−1)) :}  Δ_s =(1+(√2))^3 (1−(√2))^5  −(1+(√2))^5 (1−(√2))^3   =(1+(√2))^3 (1−(√2))^3 (1−(√2))^2  −(1+(√2))^2 (1−(√2))^3 (1+(√2))^3   =−(1−(√2))^2   +(1+(√2))^2  =3+2(√2)−(3−2(√2)) =4(√2)≠0 ⇒  α =(Δ_α /(Δs)) =( determinant (((o            (1−(√2))^3 )),((−1             (1−(√2))^5 )))/(4(√2))) =(((1−(√2))^3 )/(4(√2)))  β =(Δ_β /Δ_s )  =( determinant ((((1+(√2))^3           0)),(((1+(√2))^5           −1)))/(4(√2))) =−(((1+(√2))^3 )/(4(√2)))  ⇒ a_n =(((1−(√2))^3 )/(4(√2)))(1+(√2))^n  −(((1+(√2))^3 )/(4(√2)))(1−(√2))^n

Commented bybemath last updated on 22/Jul/20

thank you sir

Answered by john santu last updated on 21/Jul/20

Commented bybemath last updated on 21/Jul/20

wow...

Commented bybemath last updated on 21/Jul/20

how to get A and B sir?

Answered by mr W last updated on 21/Jul/20

a_5 =2a_4 +a_3   −1=2a_4 +0 ⇒a_4 =−(1/2)  a_4 =2a_3 +a_2   −(1/2)=2×0+a_2  ⇒a_2 =−(1/2)  a_3 =2a_2 +a_1   0=2(−(1/2))+a_1  ⇒a_1 =1  a_2 =2a_1 +a_0   −(1/2)=2×1+a_0  ⇒a_0 =−(5/2)    a_(n+1) −2a_n −a_(n−1) =0  let a_n =Cλ^n   Cλ^(n+1) −2Cλ^n −Cλ^(n−1) =0  Cλ^(n−1) (λ^2 −2λ−1)=0  ⇒λ^2 −2λ−1=0  ⇒λ_1 =1+(√2)  ⇒λ_2 =1−(√2)  ⇒a_n =C_1 (1+(√2))^n +C_2 (1−(√2))^n     a_0 =C_1 +C_2 =−(5/2)  a_1 =C_1 (1+(√2))+C_2 (1−(√2))=1  ⇒C_1 =((7(√2)−10)/8)  ⇒C_2 =−((7(√2)+10)/8)    ⇒a_n =(((7(√2)−10)(1+(√2))^n −(7(√2)+10)(1−(√2))^n )/8)

Commented bybemath last updated on 22/Jul/20

thank you sir