Zeno’s Paradox

Algebra Level 2

“Zeno’s paradox of Tortoise and Achilles. Zeno of Elea is credited with creating several famous paradoxes, and perhaps the best known is the paradox of Tortoise and Achilles. ... Achilles laughed louder than ever.”

You are solving for S: S = 1 + 1 2 \frac{1}{2} + 1 4 \frac{1}{4} + 1 8 \frac{1}{8} + 1 16 \frac{1}{16} ... 1 i n f i n i t y \frac{1}{infinity}

What is S equal to?


The answer is 2.

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Fresh 750 by Fresh 750 , 18, USA

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1 solution

Fresh 750
Nov 3, 2017
  • S = 1 + 1 2 \frac{1}{2} + 1 4 \frac{1}{4} + 1 8 \frac{1}{8} + 1 16 \frac{1}{16} ... 1 i n f i n i t y \frac{1}{infinity}
  • 1 2 \frac{1}{2} S = 1 2 \frac{1}{2} + 1 4 \frac{1}{4} + 1 8 \frac{1}{8} + 1 16 \frac{1}{16} ... 1 i n f i n i t y \frac{1}{infinity}

As you can see, from here, every value of S gets subtracted by every value of 1 2 \frac{1}{2} S (less 1). That is because every value to the right of 1 on the top equation “cancels” with every value of 1 2 \frac{1}{2} S. Let’s keep going.

  • 1 2 \frac{1}{2} S = 1

  • 1 2 \frac{1}{2} S(2) = S{1 x 2}

  • S = 2

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