Question

In how many ways can you climb a ladder with \(n\) rungs if at each step you can go up either one or two rungs? The terms of a sequence are given recursively as \(a_{1}=1 .\) \(a_{2}=2,\) and \(a_{n}=a_{n-1}+a_{n-2}\) for \(n \geq 2 .\) Prove by induction that \(b_{n}=F_{n+1}\) gives the terms of this sequence where \(F_{n+1}\) is the \((n+1)\) st Fibonacci number.

Short answer

The number of ways is given by the \((n+1)\)-th Fibonacci number, \(F_{n+1}\).

Step-by-step solution

Understanding the Sequence

First, we need to understand that the sequence is based on a recursion relation, where each term represents the number of ways to climb a ladder with the specified number of rungs using the allowed steps.

Base Case Verification

Verify the base cases for the sequence: \(a_1=1\) and \(a_2=2\). The first Fibonacci numbers are \(F_2=1\) and \(F_3=2\), which match the initial sequence terms.

Inductive Hypothesis

Assume that for some integer \(k\), the formula \(b_k = F_{k+1}\) holds true, i.e., the number of ways to climb a ladder with \(k\) rungs corresponds to the \((k+1)\)-th Fibonacci number.

Inductive Step

We need to prove that if \(b_k = F_{k+1}\) and \(b_{k+1} = F_{k+2}\), then \(b_{k+2} = F_{k+3}\). We know from the sequence definition that \(b_{k+2} = b_{k+1} + b_k\), and from the Fibonacci property \(F_{n+2} = F_{n+1} + F_n\). Assume \(b_k = F_{k+1}\) and \(b_{k+1} = F_{k+2}\), hence \(b_{k+2} = F_{k+2} + F_{k+1} = F_{k+3}\).

Conclusion

By the principle of mathematical induction, the statement is true for all \(n\geq1\): The number of ways to climb a ladder with \(n\) rungs corresponds to the \((n+1)\)-th Fibonacci number.

Key concepts

Mathematical Induction

Mathematical induction is a powerful technique used to prove statements or formulas that hold for a set of natural numbers. The process involves two key steps:

• **Base Case**: Verify the statement for the initial value, usually denoted as the smallest natural number involved, commonly 1 or 0.
• **Inductive Step**: Assume the statement is true for some arbitrary natural number \(k\) (this assumption is the inductive hypothesis), and then prove that its truth for \(k\) implies its truth for \(k + 1\).

This method hinges on the principle that if the base case is true, and each case leads to the next, then all subsequent cases must be true as well. In our ladder problem, the base case checks the number of ways to climb one or two rungs. We find they match the initial Fibonacci numbers exactly. The inductive step uses this base to show consistency across all numbers, following the Fibonacci sequence.

Recursive Sequence

A recursive sequence is one where each term is defined in terms of previous terms in the sequence. This type of sequence is a key aspect of the Fibonacci sequence, where each number is the sum of the two preceding ones.

For a recursive sequence, you need:

• A **base case**: Initial terms of the sequence that kick-start the recursion. For our ladder problem, they are \(a_1=1\) and \(a_2=2\).
• A **recurrence relation**: This is a rule that relates each term to its predecessors. Here, it's \(a_{n}=a_{n-1}+a_{n-2}\), mirroring the Fibonacci definition \(F_{n}=F_{n-1}+F_{n-2}\).

The recursive sequence allows solving complex problems like how to climb a ladder with variable steps. It simplifies direct computation by building results from earlier calculations, effectively leveraging previous work for new outcomes.

Combinatorial Counting

Combinatorial counting involves determining how many different ways a specific task can be performed. In our exercise, this task is climbing a ladder. The rule allows stepping either one rung or two at a time.

To formulate this, you can think of each rung in terms of smaller steps, with possibilities expanding akin to branching paths in a tree.

Using the Fibonacci sequence (**a famous tool in combinatorics**), we recognize each number as a sum of its two precedents, analogous to the choices in climbing:

• **One-step move**: Continue directly from the previous rung.
• **Two-step move**: Jump from two rungs down.

Combinatorial counting with recursive sequences showcases how tasks with simple choices, like stepping on a ladder, align perfectly with the Fibonacci numbers. This not only ensures that we count each path correctly but also highlights the sequence's versatility in real-world applications.