Grains of Rice on a Chessboard

The story is told of an Indian Prince so pleased with a wise man, who designed the game of chess, that he offered the wise man a reward of his own choice.
When the wise man chose to take the rice that could be placed on a chessboard - 64 squares - according the the following rule:
One grain on the first square, two gains on the second, four grains on the third, and so on, with the number of grains doubling with each square, the prince was surprised at what seemed so humble a request.
The number of grains of rice on the chessboard according to this rule is:
\[1+2+2^2+2^3+...+2%{63}\]

This is a geometric series with first term  
\[a=1\]
  and common ratio  
\[r=2\]
. There are  
\[n=64\]
  terms. The total number of grains of rice is found from the formula for the sum of a geometric series,  
\[S_n = \frac{a(1-r^n)}{1-r}\]
.
\[S_{64}= \frac{1 \times (1-2^{64})}{1-2}=2^{64}-1 \simeq 1.84 \times 10^{19}\]

Please don't try and count this many grains of rice. It would be easier to weigh them. The weight would be about  
\[2.88 \times 10^{11}\]
  tonnes. (1 tonne = 1000 kg).

Add comment

Security code
Refresh