Is the Sum of Coefficients of a Continued Fraction unique?

Question

Let $a$ be a rational number and $$ a = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\ddots}} \iff a = [a_0,a_1,\ldots,a_i] $$ a corresponding continued fraction. Now, the coefficients of $a$, i.e $a_0,a_1,\ldots$ are not unique. Take for example $$ 2 = [2] \quad \text{or} \quad 2 = [1,1]. $$ But it appears to me, that the sum of the coefficients $a_0+a_1+\ldots+a_i$ is indeed unique. Is this true? How can it be proven?

Answer 1

I have understood this question to mean that, for example, there is only one way of expressing $\frac{1}{2}$ as a fraction but there are two continued fraction expressions of $\frac{1}{2}$, namely, [0; 2] and [0; 1, 1]. The question is (I think) why this is the case. My answer is that it is not actually the case. For example: $\frac{1}{2}$ can be written as a continued fraction $\frac{1}{1+\frac{1}{1}}$ and generally $\frac{p}{q}$ can be written as $\frac{p}{(q-1)+\frac{1}{1}}$ . This is an unusual and generally unnecessary way of writing a continued fraction but it can be useful when considering, for example, the ordering of continued fractions and reversal of continued fractions.