Alabama State College of Barber Styling Discrete Structure Questions

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#1. Prove that n is odd if and only if
n3 is odd for all n E N .
#2. Prove that
1
f,, , 1 > ( +/
5J_
,for all n > I where f denotes the Fibonacci sequence.
#3. Suppose that f is a recursively defined function from z +to z +such that /(1) =1and /(2) = 5
and /(n+l)= /(n)+2/(n-l)for all n > 2. Prove that /(n) =2n +(-l)n.
#4. Prove that
f,(2i) 2 = (2n )(2n + 1)(2n + 2)
i=l
6
#5. Let
a,b,c,d e
JR
such that
ad-be ,o Oand c,. 0.
Define f:
JR-{-: } ➔ JR-{:} by
f (x) = ax+ b . Prove that f (x) is injective and surjective and calculate f -1 (x).
cx+d
#6. Define the sequence
n >2.
zn = (2 + n }3n for all n > 0 .
Prove that { zn } satisfies
zn = 6zn-I -9zn_2
for all

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