Nested Fraction

Algebra Level 3

Given: a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 a 4 = 222 155 a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{a_4}}}}=\frac{222}{155}

Evaluate: 1 0 4 a 0 + 1 0 3 a 1 + 1 0 2 a 2 + 1 0 1 a 3 + 1 0 0 a 4 = ? 10^4a_0+10^3a_1+10^2a_2+10^1a_3+10^0a_4 =? where a 0 , a 1 , a 2 , a 3 , a 4 , a 5 a_0,a_1,a_2,a_3,a_4,a_5 are positive integers.

14235 13452 15432 12354 12345

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Ravneet Singh
Jul 26, 2017

222 155 = 1 + 67 155 = 1 + 1 155 67 = 1 + 1 2 + 21 67 = 1 + 1 2 + 1 67 21 = 1 + 1 2 + 1 3 + 4 21 \dfrac{222}{155} = 1 + \dfrac{67}{155} = 1 + \cfrac{1}{\dfrac{155}{67}} = 1 + \cfrac{1}{2 + \dfrac{21}{67}} = 1 + \cfrac{1}{2 + \dfrac{1}{\dfrac{67}{21}}} = 1 + \cfrac{1}{2 + \dfrac{1}{3 +\dfrac{4}{21}}}

= 1 + 1 2 + 1 3 + 1 21 4 = 1 + 1 2 + 1 3 + 1 5 + 1 4 = 1 + \cfrac{1}{2 + \dfrac{1}{3 +\dfrac{1}{\dfrac{21}{4}}}} = 1 + \cfrac{1}{2 + \dfrac{1}{3 +\dfrac{1}{5 + \dfrac{1}{4}}}}

Here a 0 = 1 , a 1 = 2 , a 2 = 3 , a 3 = 5 , a 4 = 4 a_0 = 1, a_1 = 2, a_2 = 3, a_3 = 5, a_4 = 4

So, 1 0 4 a 0 + 1 0 3 a 1 + 1 0 2 a 2 + 1 0 1 a 3 + 1 0 0 a 4 = 12354 10^4a_0+10^3a_1+10^2a_2+10^1a_3+10^0a_4 = \boxed{12354}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you for sharing a nice solution.

Hana Wehbi - 3 years, 10 months ago

Relevant wiki: https://brilliant.org/wiki/continued-fractions/

Alex Li - 3 years, 10 months ago

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