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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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