One Minus A Colorful Sum

Algebra Level 2

Simplify the colorful expression below. 1 − ( 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + 1 128 + 1 256 ) {1-\left(\dfrac{1}{\color{#3D99F6}{2}}+\dfrac{1}{\color{#D61F06}{4}}+\dfrac{1}{\color{#20A900}{8}}+\dfrac{1}{\color{#624F41}{16}}+\dfrac{1}{\color{magenta}{32}}+\dfrac{1}{\color{#EC7300}{64}}+\dfrac{1}{\color{#69047E}{128}}+\dfrac{1}{\color{#BBBBBB}{256}}\right)}

0 1 256 \frac{1}{256} 1 2 1 16 \frac{1}{16} 1 128 \frac{1}{128}

Siva Prasad by Siva prasad , 20, India

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3 solutions

Nowras Otmen
Oct 20, 2015

x = 1 − ∑ n = 1 8 ( 2 − 1 ) n x=1-\displaystyle\sum_{n=1}^8 \left(2^{-1}\right)^{n}

∑ n = 1 8 ( 2 − 1 ) n = 2 − 1 ( 2 − 8 − 1 ) − 2 − 1 \displaystyle\sum_{n=1}^8 \left(2^{-1} \right)^{n} = \frac {2^{-1}(2^{-8}-1)}{-2^{-1}}
= − ( 2 − 8 − 1 ) =-\left(2^{-8}-1\right) = − 1 256 + 1 =-\frac {1}{256}+1

Making the substitution:

x = 1 − ( − 1 256 + 1 ) x=1-\left(-\dfrac {1}{256}+1\right) x = 1 + 1 256 − 1 x=1+\dfrac {1}{256}-1 Therefore: x = 1 256 x=\dfrac{1}{256}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

1 − ( 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + 1 128 + 1 256 ) = 1 − 2 7 + 2 6 + 2 5 + 2 4 + 2 3 + 2 2 + 1 256 = 1 − 2 8 − 1 256 = 1 − 255 256 = 256 − 255 256 = 1 256 1-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}\right) \\=1-\frac{2^7+2^6+2^5+2^4+2^3+2^2+1}{256}\\=1-\frac{2^8-1}{256}\\=1-\frac{255}{256}=\frac{256-255}{256}=\boxed{\dfrac{1}{256}}

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Achille 'Gilles'
Oct 25, 2015

3 Helpful 0 Interesting 0 Brilliant 0 Confused

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