Section 8*: Problem 4 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

The Fibonacci numbers of number theory are defined recursively by the formula $\begin{aligned} & \lambda_{1}=\lambda_{2}=1,\\ & \lambda_{n}=\lambda_{n-1}+\lambda_{n-2} & & \mbox{for }n>2. \end{aligned}$ Define them rigorously by use of Theorem 8.4.

Let $a_{0}=1$ , $\rho(f:\{1\}\rightarrow\mathbb{Z}_{+})=1$ and for $n>1$ , $\rho(f:\{1,\ldots,n\}\rightarrow\mathbb{Z}_{+})=f(n-1)+f(n)$ , then $\lambda_{1}\overset{def}{=}h(1)=1$ , $\lambda_{2}\overset{def}{=}h(2)=\rho(h|\{1\})=1$ , and for $n>2$ , $\lambda_{n}\overset{def}{=}h(n)=\rho(h|\{1,...,n-1\})=h(n-2)+h(n-1)\overset{def}{=}\lambda_{n-2}+\lambda_{n-1}$ .

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Section 8*: Problem 4 Solution

Section 8*: Problem 4 Solution