Four More Years, Four More Years

Algebra Level 1

If 2 2020 − 2 2019 − 2 2018 + 2 2017 = k × 2 2017 {2}^{{2020}}-{2}^{{2019}}-{2}^{{2018}}+{2}^{{2017}} = \ {k}\times {2}^{{2017}} , find k k .

1 3 2 4 2016

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Anish Harsha by Anish Harsha , 20, India

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2 solutions

Rishabh Jain
Feb 1, 2016

2 2020 − 2 2019 − 2 2018 + 2 2017 \Large \color{#D61F06}{2}^{\color{#20A900}{2020}}-\color{#D61F06}{2}^{\color{#3D99F6}{2019}}-\color{#D61F06}{2}^{\color{#20A900}{2018}}+\color{#D61F06}{2}^{\color{magenta}{2017}} = 2 2017 ( 2 3 − 2 1 − 2 2 + 2 0 ) = 3 × 2 2017 \Large =\color{#D61F06}{2}^{\color{magenta}{2017}} ( \color{#D61F06}{2}^{\color{#20A900}{3}}-\color{#D61F06}{2}^{\color{#3D99F6}{1}}-\color{#D61F06}{2}^{\color{#20A900}{2}}+\color{#D61F06}{2}^{\color{magenta}{0}}) \\ = \huge \color{#EC7300}{3}\times \color{#D61F06}{2}^{\color{magenta}{2017}} \ ∴ k = 3 \huge \therefore \huge \color{#624F41}{\boxed {\color{#EC7300}{k} =\color{darkviolet}{3}}}

15 Helpful 0 Interesting 1 Brilliant 0 Confused

⇒ 2 2020 − 2 2019 − 2 2018 + 2 2017 = k × 2 2017 \Large \color{grey}{\Rightarrow} \large \color{#D61F06}{2}^{\color{#20A900}{2020}}-\color{#D61F06}{2}^{\color{#3D99F6}{2019}}-\color{#D61F06}{2}^{\color{#20A900}{2018}}+\color{#D61F06}{2}^{\color{magenta}{2017}} = \ \color{#EC7300}{k}\times \color{#D61F06}{2}^{\color{#302B94}{2017}}

2 2017 ( 2 3 − 2 2 − 2 1 + 2 0 ) 2 2017 = k \Large \frac{\color{#D61F06}{2}^{\color{#302B94}{2017}}(\color{#D61F06}{2}^\color{#20A900}{3}-\color{#D61F06}{2}^\color{#3D99F6}{2}-\color{#D61F06}{2}^\color{#20A900}{1}+\color{#D61F06}{2}^\color{magenta}{0})}{\color{#D61F06}{2}^{\color{#302B94}{2017}}}=\color{#EC7300}{k}

2 3 − 2 2 − 2 + 1 = k \Large \color{#D61F06}{2}^\color{#20A900}{3}-\color{#D61F06}{2}^\color{#3D99F6}{2}-\color{#D61F06}{2}+\color{magenta}{1}=\color{#EC7300}{k}

8 − 4 − 2 + 1 = k \Large \color{#20A900}{8}-\color{#3D99F6}{4}-\color{#D61F06}{2}+\color{magenta}{1}=\color{#EC7300}{k}

∴ k = 3 \Large \color{#BA33D6}{\therefore} \color{#EC7300}{k}=\boxed{\color{#624F41}{3}}

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Really colourful !!

Akshat Sharda - 5 years, 4 months ago

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THANKS. \large \color{#302B94} {\text{THANKS.}}

A Former Brilliant Member - 5 years, 4 months ago

I just say "Oops!".

Prasit Sarapee - 5 years, 4 months ago

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