Question 73
The Fibonacci sequence starts with $F_{0}=0, F_{1}=1$ and, from the second term, each new term is obtained by adding the previous two, that is, $$
F_{n+2}=F_{n+1}+F_{n} \text{ to } n \geq 0
$$ Thus, the first terms of the Fibonacci sequence are: $$
\begin{array}{ccccccccccccc}
F_{0} & F_{1} & F_{2} & F_{3} & F_{4} & F_{5} & F_{6} & F_{7} & F_{8} & F_{9} & F_{10} & F_{11} & F_{12} \\
0 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 & 144
\end{array}
$$
a) Check that $F_{n+3}\lt 5 F_{n}$ for all $n \geq 3$.
b) is $n$ a positive integer. Show that between consecutive powers of $n$ there is at most $n$ Fibonacci numbers.
F_{n+2}=F_{n+1}+F_{n} \text{ to } n \geq 0
$$ Thus, the first terms of the Fibonacci sequence are: $$
\begin{array}{ccccccccccccc}
F_{0} & F_{1} & F_{2} & F_{3} & F_{4} & F_{5} & F_{6} & F_{7} & F_{8} & F_{9} & F_{10} & F_{11} & F_{12} \\
0 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 & 144
\end{array}
$$
a) Check that $F_{n+3}\lt 5 F_{n}$ for all $n \geq 3$.
b) is $n$ a positive integer. Show that between consecutive powers of $n$ there is at most $n$ Fibonacci numbers.