Answer
The terms $a,b,c$ in the simplified expressions $\dfrac{ax+b}{bx+c}$ are the terms of Fibonnaci sequence, where $a=F_{n+1}, b=F_{n}, c=F_{n-1}$ [Here, $F_n$ represents the $n^{th}$ term of the Fibonnaci sequence defined by $F_0=0, F_1=1$ and $F_n=F_{n-1}+F_{n-2}$, $n\ge2$].
Work Step by Step
Let's simplify the sequence of continued fractions and compare with the form $\dfrac{ax+b}{bx+c}$.
The Sequence of the continued fractions can be simplified as follows:
1)
$1+\dfrac{1}{x}=\dfrac{x+1}{x}=\dfrac{1\cdot x+1}{1\cdot x+0}$, where $a=1,b=1,c=0$
2)
$1+\dfrac{1}{1+\dfrac{1}{x}}=1+\dfrac{1}{\dfrac{x+1}{x}}=1+\dfrac{x}{x+1}=\dfrac{2x+1}{x+1}$, where $a=2,b=1,c=1$.
3)
$1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{x}}}=1+\dfrac{1}{\dfrac{2x+1}{x+1}}=1+\dfrac{x+1}{2x+1}=\dfrac{3x+2}{2x+1}$
, where $a=3,b=2,c=1$.
4)
$1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{x}}}}=1+\dfrac{1}{\dfrac{3x+2}{2x+1}}=1+\dfrac{2x+1}{3x+2}=\dfrac{5x+3}{3x+2}$
, where $a=5, b=3,c=2$
Clearly, the terms $a,b,c$ are the three consecutive terms of the Fibonacci Sequence.
Thus, the terms $a,b,c$ in the simplified expressions $\dfrac{ax+b}{bx+c}$ are the terms of Fibonnaci sequence, where $a=F_{n+1}, b=F_{n}, c=F_{n-1}$ [Here, $F_n$ represents the $n^{th}$ term of the Fibonnaci sequence defined by $F_0=0, F_1=1$ and $F_n=F_{n-1}+F_{n-2}$, $n\ge2$].
