ScholarMatic: Explanation & Answer
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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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ScholarMatic: Explanation & Answer
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