Continued fractions – 2

Algebra Level 3

Does there exist a real number x x with decimal representation x = 0. a 1 a 2 a 3 … ‾ x = 0.\overline{a_1 a_2 a_3 \ldots} (can be rational or irrational) that can be written as the continued fraction below?

x = 1 a 1 + 1 a 2 + 1 a 3 + 1 ⋱ \large x = \frac 1 {a_1 + \frac 1 {a_2 + \frac 1 {a_3 + \frac 1 {{}_\ddots}}}}

No Yes

Henry U by Henry U , 17, Germany

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1 solution

Alapan Das
Dec 17, 2018

I will take only three terms and try to check it . Let there are a,b and c.Again let x=.abcabcabc.....=(a+b/10+c/100)/9.Now the given term can be simplified to y=(bc+1)/(abc+a+c) which is less than x because numaretor of y lags a.If we decrease a then x will decrease and y will increase.So at one point (a) they will meet but that's a fractional value.So we are done logically for three in tigers a,b,c.And these holds for more intigers.

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