In this exercise we use the result in Exercise 74 to derive the following formula for

Chapter 13, Problem 75

(choose chapter or problem)

In this exercise we use the result in Exercise 74 to derive the following formula for the nth Fibonacci number: (1) The clever method used here was discovered by Erwin Just; it appeared in Mathematics Magazine, vol. 44 (1971), p. 199. Let a and b denote the roots of the quadratic equation x2 x 1. Then, according to Exercise 74, for n 2 we have (2) and (3) (a) Subtract equation (3) from equation (2) to show that F (4)(b) Use the quadratic formula to show that the roots ofthe equation x2 x 1 are given by a (1and b (1 (c) In equation (4), substitute for a and b using the valuesobtained in part (b). Show that this leads toequation (1).(d) The work in parts (a) through (c) shows that equation(1) holds for n 2. Now complete the derivation bychecking that equation (1) also holds for n 1.7