a-1-b-1-c-1-d-2-1-3-If-a-b-c-d-are-positive-integers-then-what-is-the-value-of-b-

Question Number 113006 by Aina Samuel Temidayo last updated on 10/Sep/20

a + \frac{1}{b + \frac{1}{c + \frac{1}{d + \dots}}} = \sqrt[3]{2}

If a, b, c, d, \dots are positive integers, then

what is the value of^{'} b^{'} ?

Answered by mr W last updated on 11/Sep/20

\sqrt[3]{2} > 1

\Rightarrow a = 1

\frac{1}{b + \dots} = \sqrt[3]{2} - 1 > \frac{1}{4}

\Rightarrow b + \dots < 4

\Rightarrow b ⩽ 3

\frac{1}{b + \dots} = \sqrt[3]{2} - 1 < \frac{1}{3}

\Rightarrow b + \dots > 3

\Rightarrow b ⩾ 3

\Rightarrow b = 3

Commented by Aina Samuel Temidayo last updated on 11/Sep/20

Why is a = 1 ?

Commented by mr W last updated on 11/Sep/20

a + \frac{1}{b + \dots} = \sqrt[3]{2} > 1, < 2

\Rightarrow a = 1

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