设数列{An},{Bn}定义如下：.
1个回答
(n+1) = 2bn +3b(n-1)
The aux.equation
x^2-2x-3 =0
x = 3 or -1
let
bn = A3^n + B(-1)^n
b0 = A+B = 1 (1)
b1= 3A-B= 7 (2)
(1)+(2)
A=2
B=-1
bn = 2.3^n -(-1)^n
a(n+1) = an+2an
The aux.equation
x^2-x-2=0
x= 2 or -1
let
an = C.2^n +D(-1)^n
a0= C+D =1 (3)
a1=2C-D =1 (4)
(3)+(4)
C=2/3
D= 1/3
an = (2/3).2^n + (1/3)(-1)^n
an = bn
(2/3).2^n + (1/3)(-1)^n= 2.3^n -(-1)^n
2.3^n - (2/3).2^n = (4/3) (-1)^(n+1)
let
f(x) = 2.3^x - (2/3).2^x
f'(x) = 2.ln3 .3^x - (2/3) ln2 .2^x =0
2.ln3 .3^x = (2/3) ln2 .2^x
(2/3)^x = 3ln3/ln2
x = ln(3ln3/ln2)/ ln(2/3)
=-0.63
f(n) = 2.3^n - (2/3).2^n
min f(n) = f(0) = 2 -2/3 = 4/3
=> f(n)> 4/3 ( n>0)
(4/3) (-1)^(n+1) = 4/3 if n=1,3,5,..
= -4/3 if n=0,2,4,6,...
2.3^n - (2/3).2^n = (4/3) (-1)^(n+1)
only when n=0
除了a0 =b0,这两个数列没有其他相同
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