If x_{1}=-1,\ x_{2}=1,\ X_{n}=3X_{(n-1)}-|2X_{(n-2)},\ \forall n\geq3. Find the gene

remolatg 2021-08-22 Answered
If x1=1, x2=1, Xn=3X(n1)2X(n2), n3. Find the general term Xn
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Expert Answer

l1koV
Answered 2021-08-23 Author has 100 answers
Step 1
The given recurrence relation is
xn3xn1+2xn2=0
The characteristic equation is
r23r+2=0(r1)(r2)=0
has the roots r1=1 and r2=2
Step 2
Since the roots are distinct, the general solution is
xn=c1r1n+c2r2n
=c11n+c22n
=c1+c22n
Step 3
Plug the initial conditions to find the value of constants.
1) 1=x1=c1+2c2
Also
2) 1=x2=c1+4c2
Step 4
Solving (1) and (2).
Subtract (1) from (2).
2c2=2c2=1
From (1),
c1=3
Step 5
Thus the general solution becomes
xn=3+2n
which is the general term.
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