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Question 353769: Please Please help me with this problem! I have been trying to get the last answer for about a week and have not been able to figure it out for the life of me. What am I missing? I have tried the third problem so many times that I am not even sure what I originally thought the answer was. I was able to figure out the first, second, and fourth answers but not the third.
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. How does it go on? Compute the next two values of f(n): f(4) =___ ~ f(5) =___ ~
Ponder the structure of this summation and then enter an algebraic expression that defines
f(n) =____ as a function of n. 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 ____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.
Here are the answers I have so far:
1. 15
2. 31
3. ?
4. 61.4819
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. How does it go on?
Compute the next two values of f(n):
f(4) = 2^4-1 = 15___ ~ f(5) = 2^5-1 = 31___
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Ponder the structure of this summation and then enter an algebraic expression that defines
f(n) =2^n-1____ as a function of n.
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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 ____tons of rice for every person on the planet.
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# of grains = 2^64-1
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(2^64-1)grains/25000(grains/lb)/2000(lbs/ton)/6 million people
= 1,537,228,673(tons per person)
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Cheers,
Stan H.
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