Further to the previous post in which he spoke of progressions, today we focus on the geometric progression, which follow an exponential growth, so that the terms of the inordinately trigger progression (for high) and vanishes quickly in the event (low). 
influenza virus A clear example of exponential growth is the spread of viruses such as influenza, leading to a large extent number of people on the face of the planet, in a few weeks, as well as other natural events ...
But today I will tell you the famous legend of the inventor of chess, reads history, the Persian king dead boring at times, suddenly became fascinated by the game of chess which he / herself a witty and clever inventor. It was so grateful that the king offered the East mathematical whatever he wanted.
The inventor replied
- I'll settle for 1 grain of wheat for the first board box, 2 for second, 4 for third, 8 for fourth and so on until the box 64 of the board.
(ie the sum of the 64 first terms of a ratio PG 2 and whose first term is 1)
King scoffed thinking it was the minutiae asking and asking his vizier to prepare the award requested, did the math and realized it was impossible to enforce the order, as the sum of the grains of the 64 cells was nothing less than the amount of:
18,446 616 grains
.744.073.709.551 (In every kilogram of wheat fit approximately 28 220 grains, so that the result would be about 653 676 260 585 tonnes, that would occupy a cube-shaped deposit of just over 11'5 kilometers on a side.
To produce such a quantity of wheat would need to be cultivating the earth (including oceans), for eight years)
To produce such a quantity of wheat would need to be cultivating the earth (including oceans), for eight years)
A second part of the story, which is next due to the embarrassment of King of having to accept that there was enough grain to pay, check with other intelligent and witty man of his court for him to pull out of trouble.
And this he proposed that:
- the inventor to see how generous you are, offer not only the sum of the 64 first terms, but the infinite sum.
To which the king said:
- You're crazy!. If I have to pay as I do to extend the sum to infinity would be infinite grains ...
ingenious But the assistant said
- Take me to the inventor and trust me, everything will be alright!
Once assembled, the proposed to assistant ingenious inventor, that the king was so pleased and happy with the game of chess and was so generous, not only offered to give the sum of 64 squares, but the infinite sum. To which, the inventor shrug accepted. And the king's aide went on to explain: Let's call
And this he proposed that:
- the inventor to see how generous you are, offer not only the sum of the 64 first terms, but the infinite sum.
To which the king said:
- You're crazy!. If I have to pay as I do to extend the sum to infinity would be infinite grains ...
ingenious But the assistant said
- Take me to the inventor and trust me, everything will be alright!
Once assembled, the proposed to assistant ingenious inventor, that the king was so pleased and happy with the game of chess and was so generous, not only offered to give the sum of 64 squares, but the infinite sum. To which, the inventor shrug accepted. And the king's aide went on to explain: Let's call S = 1 + 2 + 4 + 8 + 16 + ... (An infinite sum)
now multiply by 2, so we have 2S,
Then we 2S - 1S,
2S = 2 + 4 + 8 + 16 + 32 + ...
Then we 2S - 1S,
2S = 2 + 4 + 8 + 16 + 32 + ...
-1S = - 1 - 2 - 4 - 8 to 16 - ... ______________________
S = -1
S = -1
So we see that 2 and -2 are canceled, and 4 and -4, and equal to infinity ... so that in the end, S = -1. Not only does he no longer had to pay the inventor, but above this it was a pimple. Amazing! (So \u200b\u200bis it about playing with the infinite, such cases are called paradoxes of the infinite )
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