Inspired by MIT Problem (II)

Algebra Level 3

Solve the equation: x + 4 x + 16 x + ⋯ + 4 2018 x + 3 − x = 1 \sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\dots+\sqrt{4^{2018}x+3}}}}}-\sqrt{x}=1

1 2 2017 \frac{1}{2^{2017}} 1 2 2018 \frac{1}{2^{2018}} 1 2 4036 \frac{1}{2^{4036}} 1 2 2036 \frac{1}{2^{2036}} 1 2 2019 \frac{1}{2^{2019}}

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

x + 4 x + 16 x + ⋯ + 4 2018 x + 3 − x = 1 x + 4 x + 16 x + ⋯ + 4 2018 x + 3 = x + 1 Squaring both sides x + 4 x + 16 x + 64 x + ⋯ + 4 2018 x + 3 = x + 2 x + 1 4 x + 16 x + 64 x + ⋯ + 4 2018 x + 3 = 2 x + 1 Squaring both sides 16 x + 64 x + 256 x + ⋯ + 4 2018 x + 3 = 4 x + 1 Squaring both sides ⋯ ⟹ 4 2018 x + 3 = 2 2018 x + 1 Squaring both sides 4 2018 x + 3 = 4 2018 x + 2 ⋅ 2 2018 x + 1 ⟹ x = 1 2 2018 x = 1 2 4036 \begin{aligned} \sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\cdots + \sqrt{4^{2018}x+3}}}}} - \sqrt x & = 1 \\ \sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\cdots + \sqrt{4^{2018}x+3}}}}} & = \sqrt x + 1 & \small \color{#3D99F6} \text{Squaring both sides} \\ x + \sqrt{4x+\sqrt{16x+\sqrt{64x+\sqrt{\cdots + \sqrt{4^{2018}x+3}}}}} & = x + 2\sqrt x + 1 \\ \sqrt{4x+\sqrt{16x+\sqrt{64x+\sqrt{\cdots + \sqrt{4^{2018}x+3}}}}} & = 2\sqrt x + 1 & \small \color{#3D99F6} \text{Squaring both sides} \\ \sqrt{16x+\sqrt{64x+\sqrt{256x+\sqrt{\cdots + \sqrt{4^{2018}x+3}}}}} & = 4\sqrt x + 1 & \small \color{#3D99F6} \text{Squaring both sides} \\ \cdots \implies \sqrt{4^{2018}x+3} & = 2^{2018}\sqrt x + 1 & \small \color{#3D99F6} \text{Squaring both sides} \\ 4^{2018}x+3 & = 4^{2018}x + 2\cdot 2^{2018} \sqrt x + 1 \\ \implies \sqrt x & = \frac 1{2^{2018}} \\ x & = \boxed{\dfrac 1{2^{4036}}} \end{aligned}

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Thank You Sir.

Hana Wehbi - 2 years, 8 months ago

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