Desenhe um primeiro quadrado e pinte-o de verde. Conecte os pontos médios dos seus lados para desenhar um segundo quadrado e pinte-o de amarelo. Conecte os pontos médios do segundo quadrado para desenhar um terceiro quadrado e pinte-o de verde. Então, repita o processo para desenhar um quarto quadrado e pinte-o de amarelo. Se você continuar fazendo isto infinitamente, quais serão as frações do quadrado original pintadas em verde e em amarelo?
(Draw a first square and paint it green. Connect the midpoints of its sides to draw a second square and paint it yellow. Connect the midpoints of the sides of the second square to draw a third square and paint it green. Then, repeat the process to draw a fourth square and paint it yellow. If you keep doing this infinitely, what will be the fractions of the original square painted green and yellow?)
PCFilho
So, half of the original square is painted green, then half of that is painted yellow, then half of that is painted green, etc., etc.
ResponderExcluirSo, the total fraction painted yellow is the summation of (1/4)^k from k=1 to k=infinity. Since the base of the exponent is between -1 and 1, it converges and its sum is equal to (1/4)/(1-1/4), or 1/3. The total area painted green is twice that much, or 2/3.
My answer: 2/3 painted green, 1/3 painted yellow
Well done, Jake. That's it!
ExcluirAnother way of seeing it: every two steps, we paint definitively twice as much area green as yellow.
So, in the end, 2/3 of the area will be painted green, and 1/3 will be painted yellow.