What does it equal to?

Algebra Level 2

4 − 4 4 − 4 4 − 4 4 − 4 ⋯ = ? \Large 4-\frac 4{4-\frac 4{4-\frac 4{4-\frac 4{\cdots}}}} = \ ?


The answer is 2.

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2 solutions

Chew-Seong Cheong
Apr 28, 2017

x = 4 − 4 4 − 4 4 − 4 4 − 4 ⋯ = 4 − 4 x \begin{aligned} x & = 4-\frac 4{4-\frac 4{4-\frac 4{4-\frac 4{\cdots}}}} \\ & = 4 - \frac 4x \end{aligned}

⟹ x 2 − 4 x + 4 = 0 ( x − 2 ) 2 = 0 ⟹ x = 2 \begin{aligned} \implies x^2 - 4x + 4 & = 0 \\ (x-2)^2 & = 0 \\ \implies x = \boxed{2} \end{aligned}

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Call 4 4 − … \frac{4}{4-\dots} x. Then you get the equation 4 + 4 x = x 2 4+\frac{4}{x}=x^2 . Solve for x and you will find the answer is 2.

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