Let un be the nth Fibonacci number. Prove that the Euclidean algorithm takes precisely n steps to prove that gcd(un+1, un) = 1.

There are a few ways you can modify thestatement to make it true. We could prove that it takesn-1steps for alln≥2. Or we couldreindex the Fibonacci sequence sou0= 1,u1= 1, and prove the result for alln≥1. I’ll do thelatter.Proceed by induction onn. For the base case,u1= 1,u2= 2. So the Euclidean algorithm1 = 1·2 + 0takes exactly 1 step to prove gcd(u2,u1) = 1.Now suppose the Euclidean algorithm takesksteps to prove gcd(uk+1,uk) = 1 for somek≥1.Nowuk+2=uk+1+ukand 0<uk<uk+1(sinceuk+1=uk+uk-1). So performing the Euclideanalgorithm to find gcd(uk+2,uk+1), the first step of the algorithm will beuk+2=uk+1+uk.But now this reduces to finding gcd(uk+1,uk), which will take an additionalksteps by the inductionhypothesis. Hence, the Eucliean algorithm will take exactlyk+1 steps to find gcd(uk+2,uk+1).squaresolid

Step-by-step explanation:

(36) (x+6) (-10 or -2) i got this right on (ed)

Step-by-step explanation:

1 Add  to both sides of the equation. The value of  in this equation is 36.

2 Write the left side of the equation as a binomial squared. The left side of the equation becomes (x +6)2.

3 Use the square root property of equality.

4 Isolate the variable: x = -10 or -2