Re: a[n],b[n]
- To: mathgroup at smc.vnet.net
- Subject: [mg104916] Re: [mg104889] a[n],b[n]
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Fri, 13 Nov 2009 05:54:28 -0500 (EST)
- Reply-to: hanlonr at cox.net
Use RSolve
RSolve[{
a[n + 1] ==
1/4*(1 + Sqrt[5])*a[n] -
1/2*Sqrt[1/2*(5 - Sqrt[5])]*b[n],
b[n + 1] ==
1/2*Sqrt[1/2*(5 - Sqrt[5])]*a[n] +
1/4*(1 + Sqrt[5])*b[n],
a[1] == 4, b[1] == 9},
{a[n], b[n]}, n] // Simplify
{{a[n] -> (1/(1 + Sqrt[5]))*(2^(-(2*n) - 3)*
((4 + 9*I)*(6 + 2*Sqrt[5] - I*Sqrt[50 - 10*Sqrt[5]] -
I*Sqrt[10 - 2*Sqrt[5]])*(1 + Sqrt[5] +
I*Sqrt[10 - 2*Sqrt[5]])^n + (9 + 4*I)*
(Sqrt[50 - 10*Sqrt[5]] + Sqrt[10 - 2*Sqrt[5]] -
2*I*(3 + Sqrt[5]))*(1 + Sqrt[5] -
I*Sqrt[10 - 2*Sqrt[5]])^n)),
b[n] -> (1/(1 + Sqrt[5]))*(2^(-(2*n) - 3)*
((9 + 4*I)*(6 + 2*Sqrt[5] + I*Sqrt[50 - 10*Sqrt[5]] +
I*Sqrt[10 - 2*Sqrt[5]])*(1 + Sqrt[5] -
I*Sqrt[10 - 2*Sqrt[5]])^n - (4 + 9*I)*
(Sqrt[50 - 10*Sqrt[5]] + Sqrt[10 - 2*Sqrt[5]] +
2*I*(3 + Sqrt[5]))*(1 + Sqrt[5] +
I*Sqrt[10 - 2*Sqrt[5]])^n))}}
Bob Hanlon
---- ynb <wkfkh056 at yahoo.co.jp> wrote:
=============
a[n + 1] = 1/4*(1 + Sqrt[5])*a[n] - 1/2*Sqrt[1/2*(5 - Sqrt[5])]*b[n],
b[n + 1] = 1/2*Sqrt[1/2*(5 - Sqrt[5])]*a[n] + 1/4*(1 + Sqrt[5])*b[n],
a[1] = 4, b[1] = 9.
a[n]=
b[n]=