Geometric Progression

An ancient story illustrates the power of a geometric progression. This story has been retold in numerous versions and as taking place in many different locales, but the general plot is always the same. A king wishes to reward a man, and the man asks for a seemingly insignificant sum: taking a standard chessboard, he asks the king to give him one grain of rice on day one, two grains of rice on day two, and so on for 64 days. The king hastily agrees, not realizing that in order to provide the amount of rice required he will eventually bankrupt himself. How much rice did the king’s reward require? Assuming he could actually reach the final square of the board, he would be required to provide 9,223,372,036,854,775,808 grains of rice, which by one calculation could be grown only by planting the entire surface of the planet Earth with rice four times over. However, it is doubtful the king would have moved far past the middle section of the chessboard before realizing the folly of his generosity. The legend does not record whether the king was impressed or angered by this demonstration of mathematical wisdom.

Progresión Aritmética

de diferencia

𝒙ₙ₊₁= 𝒙ₙ + 𝐝
     = 𝒙₁ ＋ 𝒏⋅𝐝

Progresión Geométrica

de razón

𝒙ₙ₊₁ = 𝐫∙ 𝒙ₙ    
     = 𝐫ⁿ∙𝒙₁