Discrete mathematics If x_{1}=2, x_{n}=4X_{n-1}-4n \forall n \geq 2. Find t

ringearV 2021-08-16 Answered
Discrete mathematics
If x1=2,xn=4Xn14nn2.
Find the general term xn.
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Expert Answer

Nicole Conner
Answered 2021-08-17 Author has 97 answers
Step 1
Given recurrence relation is
xn=4xn14n, x1=2
Rewriting given recurrence relation, we have
xn4xn1=4n
Associated homogeneous recurrence relation is
xn4xn1=0
Auxiliary equation is
m4=0
m=4
Complementary solution is
xnc=C(4)n C is arbitrary constant.
Step 2
To find particular solution:
the non-homogeneous function is Q(n)=4n=0(1)n4n(1)n
particular solution will have form
xnp=A+Bn
xn1p=A+B(n1)=A+BnB
Substituting in given recurrence relation
A+Bn=4(A+BnB)4n
A+Bn4A4Bn+4B=4n
(3A+4B)+(3B)n=4n
3A+4B=0,3B=4 by comparing coefficients of 1&n.
B=43 and 3A=4B=4×43
A=169 and B=43
Hence particular solution is
xnp=A+Bn=169+43n
xnp=169+43n
Step 3
General solution is
xn=xnc+xnp
xn=C(4)n+169+43n
Given that, x1=2
2=C(4)1+169+43×1
2=4C+16+129

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