Chapter 5.6, Problem 39ES

With each step you take when climbing a staircase, you can move up either one stair or two stairs. As a result, you can climb the entire staircase taking one climb the entire staircase takimg one stair at a time, taking two at a time, or taking a combination of one-and two-stair increments. For each integer $n\ge 1$ , if the staircase consist of n stairs, let $c_n$ be the number of different ways to climb the straircae. Find a recurrenvce relation $c_1,c_2,c_3,\mathrm{....}$ Justify your answer.