infinite fraction 3

Number Theory Level 2

Consider the following fraction:

$\LARGE\frac{1+\frac{2+\frac{3+\frac{1+\frac{2+\frac{3+⋰}{10}}{10}}{10}}{10}}{10}}{10}$

The fraction can be formalized via recursive definition as $a$ where $a=\frac{1+b}{10}$ and $b=\frac{2+c}{10}$ and $c=\frac{3+a}{10}$

$a$ is a rational number and has a reduced fraction form $X/Y$ where $X$ and $Y$ are positive integers. What is $3X+3Y$ ?

1 solution

Chase Marangu
Oct 11, 2020

(very informal solution)

$a=\frac{1+\frac{2+\frac{3+a}{10}}{10}}{10}$

$a=\frac{123+a}{1000}$

$a=0.123123123...$

$a=123/999=41/333$

$X=41$

$Y=333$

$3X+3Y=123+999=1122$

It should be a= (1+b)/10 in the question, similarly with b and c

Valiant Thor - 8 months ago

Valiant Thor shoot sry I’ll fix that

chase marangu - 8 months ago