a)an=3an−1+4an−2 n≥2,a0=a1=1
Rewrite the recurrence relation an−3an−1−4an−2=0.
Now form the characteristic equation:
x2−3x−4=0x=−1 and x=4
We therefore know that the solution to the recurrence
relation will have the form:
an=a∗(−1)n+b∗4n
To find a and b , plug in n = 0 and n = 1 to get a system of two equations with two unknowns:
1=a∗(−1)0+b∗40=a+b1=a∗(−1)1+b∗41=4b−aa=53 and b=52 answer:an=53(−1)n+524n
b)an=an−2 n≥2,a0=a1=1
Rewrite the recurrence relation an−an−2=0.
Now form the characteristic equation:
x2−1=0x=−1 and x=1
We therefore know that the solution to the recurrence
relation will have the form:
an=a∗(−1)n+b∗1n
To find a and b , plug in n = 0 and n = 1 to get a system of two equations with two unknowns:
1=a∗(−1)0+b∗10=a+b1=a∗(−1)1+b∗11=b−aa=0 and b=1 answer:an=1∗1n=1
c)an=2an−1−an−2 n≥2,a0=a1=2
Rewrite the recurrence relation an−2an−1+an−2=0.
Now form the characteristic equation:
x2−2x+1=0x=1
We therefore know that the solution to the recurrence
relation will have the form:
an=a∗(1)n+b∗n∗1n=a+bn
To find a and b , plug in n = 0 and n = 1 to get a system of two equations with two unknowns:
2=a+b∗0=a2=a+b∗1=a+ba=2 and b=0 answer:an=2+0∗n=2
d)an=3an−1−3an−2 n≥3,a0=a1=1,a2=2
Rewrite the recurrence relation an−3an−1+3an−2=0.
Now form the characteristic equation:
x2−3x+3=0x=1/2(3−i3) orx=1/2(3+i3)
We therefore know that the solution to the recurrence
relation will have the form:
an=a∗(1/2(3−i3))n+b∗(1/2(3+i3))n
To find a and b , plug in n = 0 and n = 1 to get a system of two equations with two unknowns:
1=a+b1=a∗(1/2(3−i3))+b∗(1/2(3+i3))1/2(3+i3)−1=−ai3a=1/6(−3+i3)b=1/6(9−i3) answer:an=1/6(−3+i3)∗(1/2(3−i3))n+1/6(9−i3)∗(1/2(3+i3))n
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