Never Ending Fraction

Algebra Level 2

Find the positive value of:

2 3 + 5 3 ( 2 3 + 5 3 ( 2 3 + 5 3 ( . . . ) ) ) \large \frac{2}{3} + \frac{5}{3 \left(\frac{2}{3} + \frac{5}{3 \left(\frac{2}{3} + \frac{5}{3(...)}\right)}\right)}

7 2 \frac 72 5 3 \frac 53 3 \sqrt{3} 3 3 1 1 2 \sqrt{2}

Daniel Venable by Daniel Venable , 15, USA

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2 solutions

Chew-Seong Cheong
Feb 12, 2019

Similar solution with @Daniel Venable 's

Let x x be the value to be found. Then

x = 2 3 + 5 3 ( 2 3 + 5 3 ( 2 3 + 5 3 ( ⋯ ) ) ) x = 2 3 + 5 3 x Multiply both sides by 3 x 3 x 2 = 2 x + 5 Rearrange 3 x 2 − 2 x − 5 = 0 ( 3 x − 5 ) ( x + 1 ) = 0 ⟹ x = 5 3 Since x > 0 \begin{aligned} x & = \frac 23 + \frac 5{3\left(\frac 23 + \frac 5{3\left(\frac 23 + \frac 5{3(\cdots)} \right)}\right)} \\ x & = \frac 23 + \frac 5{3x} & \small \color{#3D99F6} \text{Multiply both sides by }3x \\ 3x^2 & = 2x + 5 & \small \color{#3D99F6} \text{Rearrange} \\ 3x^2 - 2x - 5 & = 0 \\ (3x-5)(x+1) & = 0 \\ \implies x & = \boxed{\frac 53} & \small \color{#3D99F6} \text{Since }x > 0 \end{aligned}

2 Helpful 1 Interesting 0 Brilliant 0 Confused
Daniel Venable
Feb 11, 2019

Set the equation to x x , ( x = 2 3 + 5 3 x ) (x = \frac{2}{3} + \frac{5}{3x})

Multiply by 3 x 3x , ( 3 x 2 = 2 x − 5 ) (3x^{2} = 2x - 5)

Use the quadratic formula to get − 1 -1 or 5 3 . \frac{5}{3}.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

No need to mention "positive value" because the negative value is not a solution.

Chew-Seong Cheong - 2 years, 4 months ago

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