Chapter 5.4, Problem 9ES

Define a sequence $a_1,a_2,a_3,\mathrm{...}$ as follows: $a_1=1,a_2=3$ , and $a_{k-1}+a_{k-2}$ for every integer $k\ge 3$ . (This sequence is known as the Lucas saqucnce.) Use strong mathematical induction to prove that $a_n\le {\left(\frac{7}{4}\right)}^n$ for every integer $n\ge 1$ .