Let (a, b, c) be a Pythagorean triple. If (a, b, c) form an arithmetic sequence,

Question

Let (a, b, c) be a Pythagorean triple. If (a, b, c) form an arithmetic sequence, show that (a, b, c) = (3k, 4k, 5k) or some positive integer k. Show the n^th Fibonacci number (f_0 = 0, f_1 = 1) satisfies the formula f_n = f_n = [(1 + squareroot 5)^n - (1 - squareroot 5)^n]/2^n squareroot. Let f: A rightarrow B, and let C subsetoforequal to A. Prove or give a counterexample for the following: a. f(AC) subsetoforequal C f (A)(C) b. f(A)(C) subsetoforequal f(AC) c. if f is injective, f(AC) = f(A) f(C) d. if f is surjective, f(AC) = B(C)

Explanation / Answer

1) Pythagorea triple means

C2 =B2 +A2

and A =a , B =a+d and C =a+ 2d

(a+2d)2 =a2 +(a+d)2

a2 +4d2 +4ad = a2 +a2 +d2 +2ad

3d2 +2ad -a2 =0

3d2 +3ad -ad -a2 = 3d(d+a) -a(d+a)

a =3d

A =3d ,B =3d +d =4d and C =3d +2d =5d

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