Question

A man must climb a flight of \(k\) steps. He always takes one or two steps at a time. Thus he can climb 3 steps in the following ways: 1,1,\(1 ; 1,2 ;\) or 2,1 . Find \(s_{k},\) the number of ways he can climb the flight of \(k\) steps. [Hint: Fibonacci.]

Short answer

The number of ways to climb \(k\) steps is \(F_{k+1}\), where \(F\) is the Fibonacci sequence.

Step-by-step solution

Understanding the Problem

The man can climb a flight of steps taking either one step or two steps at a time, and we need to find out in how many different ways he can climb a flight of exactly \(k\) steps.

Establishing a Formula

Let \(s_k\) denote the number of ways to climb \(k\) steps. Notice that the number of ways to reach the \(k^{th}\) step will be the sum of the ways to reach the \((k-1)^{th}\) step (by one step) and the \((k-2)^{th}\) step (by two steps).

Deriving the Recurrence Relation

From the understanding above, we can express \(s_k\) using the relation \(s_k = s_{k-1} + s_{k-2}\). This is similar to the Fibonacci sequence, where each term is the sum of the two preceding terms.

Base Cases

To solve this recurrence, we need initial conditions. Let's start by interpreting small values: For \(k = 0\), there is 1 way (doing nothing), and for \(k = 1\), there is also 1 way (taking one step). Therefore, \(s_0 = 1\) and \(s_1 = 1\).

Applying the Recurrence Relation

Use the base cases to compute further terms. For instance, \(s_2 = s_1 + s_0 = 1 + 1 = 2\) and so on. Hence, \(s_3 = s_2 + s_1 = 2 + 1 = 3\), exactly as described in the example.

General Solution

The problem has been reduced to finding terms of a Fibonacci-like sequence starting with \(s_0 = 1\) and \(s_1 = 1\). The number of ways to climb \(k\) steps is given by the \((k+1)^{th}\) Fibonacci number, denoted as \(F_{k+1}\).

Key concepts

Recurrence Relation

Understanding how sequences are formulated leads us to recurrence relations. These help in defining sequences where each term is a function of its preceding terms. A key aspect of recurrence relations is that they provide a mathematical formula that expresses each term in relation to others, instead of using explicit patterns.

In the exercise about climbing steps, we described how each way to climb the steps

• The way to get to the current step can be seen as a combination of reaching the previous one or two steps before.
• This forms a recurrence relation: \( s_k = s_{k-1} + s_{k-2} \).

This sequence resembles the Fibonacci sequence. Every term is made by summing the two terms that come just before it.

Such relations are widely applied in problem-solving because they simplify calculating terms for any size of the sequence.

Combinatorics

Combinatorics is a branch of mathematics that involves counting, arrangement, and combination of objects. It plays an instrumental role when dealing with problems involving possibilities and different arrangements.

In our problem:

• We see combinatorics in the different ways a person can step up to a step.
• Taking steps of 1 or 2, the question becomes how these can be combined to reach exactly \(k\) steps.

Looking at the first few step possibilities (as shown in the solution), highlights how combinations are confirmed. For 3 steps, the formations are: \(1,1,1\); \(1,2\); and \(2,1\). Calculating through recursive patterns (noted as recurrence relations) adds to the combinatorial nature.

Such knowledge is essential as combinatorics help verify that every potential combination accounts for reaching the top, ensuring no option is missed.

Step Counting Problem

Step counting problems are a fun way to explore sequences and patterns in mathematics through physical interpretation. They offer practical scenarios where mathematical techniques, like those from the Fibonacci sequence, are applied.

For this exercise, step counting shows how one can approach multiple ways to solve a problem, blending recurrence relations with logical thinking.

• We saw how using one or two steps reflects the choices faced at each literal step in climbing.
• The progression used hence becomes illustrative of choices needed for larger problems, where steps mirror options facing a broader solution space.

Step counting introduces problem solvers to sequence creation, pattern recognition, and expression application. These aspects go far beyond counting problems and have real-world applications in fields like computer science and operational research.