Chapter 11.1, Problem 32PE

The number in the sequence defined by $a_1=1,a_2=3$ , and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 3$ are referred to as Lucas numbers in honor of French mathematician Edouard Lucas $\left(1842-1891\right)$ .

a. Find the first eight Lucas numbers.

b. The formula $L_n={\left(\frac{1+\sqrt{5}}{2}\right)}^n+{\left(\frac{1-\sqrt{5}}{2}\right)}^n$ gives the $n$ th Lucas number. Use a calculator to verify this statement for $n=1,n=2$ , and $n=3$ .