Infinite Computing....

Algebra Level 3

Can you Calculate this... to infinity?

1 7 \frac{1}{7} + 2 7 2 \frac{2}{7^{2}} + 1 7 3 \frac{1}{7^{3}} + 2 7 4 \frac{2}{7^{4}} + 1 7 5 \frac{1}{7^{5}} + 2 7 6 \frac{2}{7^{6}} + 1 7 7 \frac{1}{7^{7}} .....


The answer is 0.1875.

Arpit MIshra by Arpit MIshra , 21, India

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1 solution

Arpit MIshra
Feb 21, 2015

Firstly , we must separate the sum into two parts, X and Y...

X = 1 7 \frac{1}{7} + 1 7 3 \frac{1}{7^{3}} + 1 7 5 \frac{1}{7^{5}} + ..... 1 7 2 n − 1 \frac{1}{7^{2}n -1}

Y = 2 7 2 \frac{2}{7^{2}} + 2 7 4 \frac{2}{7^{4}} + 2 7 6 \frac{2}{7^{6}} + ..... 2 7 2 n \frac{2}{7^{2}n}

X = 1 7 1 − 7 2 \frac{\frac{1}{7}}{1-7^{2}} = 7 48 \frac{7}{48}

Y = 2 7 2 1 − 1 7 2 \frac{\frac{2}{7^{2}}}{1-\frac{1}{7^{2}}} = 2 48 \frac{2}{48}

X+Y = 7 48 \frac{7}{48} + 2 48 \frac{2}{48} = 9 48 \frac{9}{48} = 3 16 \frac{3}{16}

= 0.188 \boxed{0.188}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Standard approach

Rama Devi - 5 years, 11 months ago

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