The Fibonacci sequence F1; F2;::: is defined byF1 D 1; F2 D 1; and Fn D Fn2 C Fn1 for n 3:Define T 2 L.R2/ by T .x; y/ D .y; x C y/.(a) Show that T n.0; 1/ D .Fn; FnC1/ for each positive integer n.(b) Find the eigenvalues of T.(c) Find a basis of R2 consisting of eigenvectors of T.(d) Use the solution to part (c) to compute T n.0; 1/. Conclude thatFn D 1p51 C p52n1 p52n for each positive integer n.(e) Use part (d) to conclude that for each positive integer n, theFibonacci number Fn is the integer that is closest to1p51 C p52n

1.5 Exponents and Radicals Math 1315-003 Shelley Hamilton Division with Exponents: 5^7/5^4 = 5^7-4 = 5^3 (-8) ^10/ (-8) ^5 = (-8) ^10-5 = (-8) ^5 (3c) ^9/(3c) ^3 = (3c) ^9-3 = (3c) ^6 (7*19) ^3 = (7*19) (7*19) (7*19) = 7*7*7*19*19*19 = 7^3 * 19^3 When dividing exponents, you just subtract the exponents as you can see in the examples above. Product to a power: (ab)^n = a^n * b^n Ex. (5y) ^3 = 5^3 y^3 = 125y^3 (c^2 d^3) ^4 = (c^2) ^4 (d^3) ^4 (x/2) ^6 = x^6/2^6 = x^6/64 (a^4/b^3) ^3 = a^12/b^9 When multiplying when there is an exponent in and one outside of the (). You multiply them together. Also, that exponent outside of the parenthesis, you distribute it in with the coefficient. Or every variable i