Question 933253: In a well known story the inventor of the game of chess was asked by his well pleased King what reward he desired. "Oh, not much, your majesty", the inventor responded, "just place a grain of rice on the first square of the board, 2 on the next, 4 on the next, and so on, twice as many on each square as on the preceding one. I will give this rice to the poor." (For the uninitiated, a chess board has 64 squares.) The king thought this a modest request indeed and ordered the rice to be delivered.
Let f(n) denote the number of rice grains placed on the first n squares of the board. So clearly, f(1)=1, f(2) = 1+2 = 3, f(3)= 1+ 2 + 4 =7, and so on.
Part I. Ponder the structure of this summation and then enter an algebraic expression that defines
f(n) = ? as a function of n.
Part II. Supposing that there are 25,000 grains of rice in a pound, 2000 pounds in a ton, and 6 billion people on earth, the inventor's reward would work out to approximately X tons of rice for every person on the planet. Clearly, all the rice in the kingdom would not be enough to begin to fill that request. The story has a sad ending: feeling duped, the king caused the inventor of chess to be beheaded. What is the X value?
Im stumbled, help would be appreciated.
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Let f(n) denote the number of rice grains placed on the first n squares of the board. So clearly, f(1)=1, f(2) = 1+2 = 3, f(3)= 1+ 2 + 4 =7, and so on.
Part I. Ponder the structure of this summation and then enter an algebraic expression that defines
f(n) = ? as a function of n.
f(n) = 1 + 2 + 2^2 + 2^3 + .... + 2^n
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geometric series with a = 1 and r = 2
Sum + a(r^(n+1)-1)/(r-1) = 2^65-1
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Part II. Supposing that there are 25,000 grains of rice in a pound, 2000 pounds in a ton, and 6 billion people on earth, the inventor's reward would work out to approximately X tons of rice for every person on the planet. Clearly, all the rice in the kingdom would not be enough to begin to fill that request. The story has a sad ending: feeling duped, the king caused the inventor of chess to be beheaded. What is the X value?
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X = 2^65/[25000*2000*6*10^9]
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Cheers,
Stan H.
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